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Flashcard 1436162067724

Tags
#costs #finance #investopedia
Question

Net income of a business reflects residual income remaining after all [...] costs have been paid.

Answer
explicit


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Examples of Explicit Costs Net income of a business reflects residual income remaining after all explicit costs have been paid. Explicit costs are the only costs necessary to calculate accounting profit. Expenses relating to advertising, supplies, utilities, inventory and equipment actually

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Explicit Cost Definition | Investopedia
An explicit cost is an expense that has occurred and has a clearly defined dollar amount. These expenses are incurred during business operations and are actual out-of-pocket cash outlays. The objective dollar amounts are subject to reporting. <span>Examples of Explicit Costs Net income of a business reflects residual income remaining after all explicit costs have been paid. Explicit costs are the only costs necessary to calculate accounting profit. Expenses relating to advertising, supplies, utilities, inventory and equipment actually purchased are examples of explicit costs. Although the depreciation of an asset is not an activity that can be tangibly traced, depreciation expense is an explicit cost because it relates to the cost of the underlying asset that the company owns. Explicit Costs vs. Implicit Costs Explicit costs arise based on what has actually been purchased as opposed to implicit costs that arise based on what has actually been given up other t







Flashcard 1438913269004



Tags
#4-3-the-investment-opportunity-set #cfa #cfa-level-1 #economics #has-images #microeconomics #reading-14-demand-and-supply-analysis-consumer-demand #section-5-consumer-equilibrium #study-session-4
Question
The marginal rate of substitution is the rate at which the consumer is willing to sacrifice wine for bread. Additionally, the price ratio is the rate at which she must sacrifice wine for another slice of bread. So, at equilibrium, the consumer is just willing to pay [...] that she must pay.
Answer
the opportunity cost


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tion is the rate at which the consumer is just willing to sacrifice wine for bread. Additionally, the price ratio is the rate at which she must sacrifice wine for another slice of bread. So, at equilibrium, the consumer is just willing to pay <span>the opportunity cost that she must pay.<span><body><html>







Flashcard 1439254056204

Tags
#eximbank #key-features-of-loan-guarantees #octopus #usa
Question
Loans are made by [...] and repayment of these loans is guaranteed by [...]
Answer
commercial banks

Ex-Im Bank


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Loans are made by commercial banks and repayment of these loans is guaranteed by Ex-Im Bank

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Government-Assisted Foreign Buyer Financing (Eximbank USA)
g may not be available in certain countries and certain terms for U.S. government policy reasons (for more information, see the Country Limitation Schedule posted on the Bank’s Web site, www.exim.gov, under the “Apply” section). <span>Key Features of Ex-Im Bank Loan Guarantees Loans are made by commercial banks and repayment of these loans is guaranteed by Ex-Im Bank. Guaranteed loans cover 100 percent of the principal and interest for 85 percent of the U.S. contract price. Interest rates are negotiable, and are usually floating and lower than fixed rates. Guaranteed loans are fully transferable, can be securitized and are available in certain foreign currencies. Guaranteed loans have a faster documentation process with the assistance of commercial banks. There are no U.S. vessel shipping requirements for amounts less than $20 million. Key Features of Ex-Im Bank Direct Loans Fixed-rate loans are provided directly to creditworthy foreign buyers. Direct loans supp







Flashcard 1442148912396

Tags
#cfa-level-1 #corporate-finance #reading-35-capital-budgeting #study-session-10
Question
intangible costs and benefits are often ignored because, [...].
Answer
if they are real, they should result in cash flows at some other time


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intangible costs and benefits are often ignored because, if they are real, they should result in cash flows at some other time.

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3. BASIC PRINCIPLES OF CAPITAL BUDGETING
ting relies on just a few basic principles. Capital budgeting usually uses the following assumptions: Decisions are based on cash flows. The decisions are not based on accounting concepts, such as net income. Furthermore, <span>intangible costs and benefits are often ignored because, if they are real, they should result in cash flows at some other time. Timing of cash flows is crucial. Analysts make an extraordinary effort to detail precisely when cash flows occur. Cash flows are based on opportunity cos







Flashcard 1474078051596

Tags
#cfa-level-1 #reading-22-financial-statement-analysis-intro
Question
To help improve the quality of the discussion by management, the [...] issued an exposure draft in June 2009 that proposed a framework for the preparation and presentation of management commentary.
Answer
International Accounting Standards Board (IASB)


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To help improve the quality of the discussion by management, the International Accounting Standards Board (IASB) issued an exposure draft in June 2009 that proposed a framework for the preparation and presentation of management commentary.

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3.1.6. Management Commentary or Management’s Discussion and Analysis
the financial statements, information included in the management commentary is typically unaudited. When using information from the management report, an analyst should be aware of whether the information is audited or unaudited. <span>To help improve the quality of the discussion by management, the International Accounting Standards Board (IASB) issued an exposure draft in June 2009 that proposed a framework for the preparation and presentation of management commentary. Per the exposure draft, that framework will provide guidance rather than set forth requirements in a standard. The exposure draft identifies five content elements of a “decision-useful







Flashcard 1611264298252

Question
Through which accounts are the shareholders' equity statement and the balance sheet related?
Answer
contributed capital

retained earnings


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Equity
that occurred during the accounting period in components of the stockholders' equity section of the balance sheet. For example, it includes capital transactions with owners (e.g., issuing shares) and distributions to owners (i.e., dividends). <span>The shareholders' equity section of the balance sheet lists the items in contributed capital and retained earnings on the balance sheet date. <span><body><html>







#reading-8-statistical-concepts-and-market-returns #skewness
In calculations of variance we do not know whether large deviations are likely to be positive or negative, hence the degree of symmetry in return distributions.

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#reading-9-probability-concepts
If we know both the set of all the distinct possible outcomes of a random variable and the assignment of probabilities to those outcomes—the probability distribution of the random variable—we have a complete description of the random variable, and we can assign a probability to any event that we might describe

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Flashcard 1647717780748

Tags
#reading-9-probability-concepts
Question
P(A|S1)P(S1) + P(A|S2)P(S2) +…+ P(A|Sn)P(Sn)

What does this equation say?
Answer
The probability of any event [P(A)] can be expressed as a weighted average of the probabilities of the event, given scenarios [terms such P(A | S1)]


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#reading-9-probability-concepts
The unconditional variance of EPS is the sum of two terms:

1) the expected value (probability-weighted average) of the conditional variances (parallel to the total probability rules) and

2) the variance of conditional expected values of EPS.

The second term arises because the variability in conditional expected value is a source of risk. Term 1 is σ2(EPS) = P(declining interest rate environment) σ2(EPS | declining interest rate environment) + P(stable interest rate environment) σ2(EPS | stable interest rate environment) = 0.60(0.004219) + 0.40(0.0096) = 0.006371.

Term 2 is σ2[E(EPS | interest rate environment)] = 0.60($2.4875 − $2.34)2 + 0.40($2.12 − $2.34)2 = 0.032414. Summing the two terms, unconditional variance equals 0.006371 + 0.032414 = 0.038785.

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#reading-9-probability-concepts
Regarding counting, there can never be more combinations than permutations for the same problem, because permutations take into account all possible orderings of items, whereas combinations do not.

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Subject 10. Principles of Counting
he ten stocks you are analyzing and invest $10,000 in one stock and $20,000 in another stock, how many ways can you select the stocks? Note that the order of your selection is important in this case. 10 P 2 = 10!/(10 - 2)! = 90 <span>Note that there can never be more combinations than permutations for the same problem, because permutations take into account all possible orderings of items, whereas combinations do not. <span><body><html>




For if it is true that no production of knowledge in the human sciences can ever ignore or disclaim its author's involvement as a human subject in his own circumstances, then it must also be true that for a European or American studying the Orient there can be no disclaiming the main· circumstances of his actuality: that he comes up against the Orient as a European or American first, as an individual second.

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A number of experiments have demonstrated the effectiveness of the KWM. One of the most common research designs has been to test the KWM against other methods of learning vocabulary. The KWM has been compared with learning words in context (e.g., immersion) or learning words with no given strategy. For example, McDaniel and Pressley (1989) found the KWM to be significantly more facilitative to learning over the context method. It has also been shown that the KWM is superior over systematic teaching (King-Sears et al., 1992). Additionally, this method has been shown to be an effective tool in learning many words in a short amount of time. Atkinson and Raugh (1975) found that students were able to learn 120 new words in three days (40 words a day). In sum, a good deal of research has shown that the KWM of vocabulary learning is highly effective.

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Flashcard 1652306873612

Tags
#reading-9-probability-concepts
Question
If variance is 0, this means there [...]
Answer
is no dispersion or risk.


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Expected value
Variance is a number greater than or equal to 0 because it is the sum of squared terms. If variance is 0, there is no dispersion or risk. The outcome is certain, and the quantity X is not random at all. Variance greater than 0 indicates dispersion of outcomes. Increasing variance indicates increasing dispersion, all else







Flashcard 1652310543628

Tags
#reading-9-probability-concepts
Question
The shortest explanation of n factorial is that it is the number of [...]
Answer
ways to order n objects in a row.


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The shortest explanation of n factorial is that it is the number of ways to order n objects in a row. In all the problems to which we apply this counting method, we must use up all the members of a group (sampling without replacement).







#reading-9-probability-concepts
In all the problems to which we apply n! method, we must use up all the members of a group (sampling without replacement).

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The shortest explanation of n factorial is that it is the number of ways to order n objects in a row. In all the problems to which we apply this counting method, we must use up all the members of a group (sampling without replacement).




Flashcard 1652316048652

Tags
#reading-9-probability-concepts
Question
Regarding counting, there can never be more [...] than [...] for the same problem.
Answer
combinations

permutations


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Regarding counting, there can never be more combinations than permutations for the same problem, because permutations take into account all possible orderings of items, whereas combinations do not.

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Subject 10. Principles of Counting
he ten stocks you are analyzing and invest $10,000 in one stock and $20,000 in another stock, how many ways can you select the stocks? Note that the order of your selection is important in this case. 10 P 2 = 10!/(10 - 2)! = 90 <span>Note that there can never be more combinations than permutations for the same problem, because permutations take into account all possible orderings of items, whereas combinations do not. <span><body><html>







Flashcard 1652318407948

Tags
#reading-9-probability-concepts
Question
Why can there never be more combinations than permutations for the same problem?
Answer

because permutations take into account all possible orderings of items, whereas combinations do not.


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Regarding counting, there can never be more combinations than permutations for the same problem, because permutations take into account all possible orderings of items, whereas combinations do not.

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Subject 10. Principles of Counting
he ten stocks you are analyzing and invest $10,000 in one stock and $20,000 in another stock, how many ways can you select the stocks? Note that the order of your selection is important in this case. 10 P 2 = 10!/(10 - 2)! = 90 <span>Note that there can never be more combinations than permutations for the same problem, because permutations take into account all possible orderings of items, whereas combinations do not. <span><body><html>







Flashcard 1652320242956

Tags
#reading-9-probability-concepts
Question
Do combinations take into account all possible orderings of items?
Answer
No


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Regarding counting, there can never be more combinations than permutations for the same problem, because permutations take into account all possible orderings of items, whereas combinations do not.

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Subject 10. Principles of Counting
he ten stocks you are analyzing and invest $10,000 in one stock and $20,000 in another stock, how many ways can you select the stocks? Note that the order of your selection is important in this case. 10 P 2 = 10!/(10 - 2)! = 90 <span>Note that there can never be more combinations than permutations for the same problem, because permutations take into account all possible orderings of items, whereas combinations do not. <span><body><html>







Flashcard 1652322077964

Tags
#reading-9-probability-concepts
Question
Do permutations take into account all possible orderings of items?
Answer
Yes


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Regarding counting, there can never be more combinations than permutations for the same problem, because permutations take into account all possible orderings of items, whereas combinations do not.

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Subject 10. Principles of Counting
he ten stocks you are analyzing and invest $10,000 in one stock and $20,000 in another stock, how many ways can you select the stocks? Note that the order of your selection is important in this case. 10 P 2 = 10!/(10 - 2)! = 90 <span>Note that there can never be more combinations than permutations for the same problem, because permutations take into account all possible orderings of items, whereas combinations do not. <span><body><html>







Flashcard 1652323912972

Tags
#probability
Question
[...] states that for a normal distribution, nearly all of the data will fall within three standard deviations of the mean.
Answer
The empirical rule


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The empirical rule states that for a normal distribution, nearly all of the data will fall within three standard deviations of the mean. The empirical rule can be broken down into three parts: 68% of data

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the+empirical+rule - Buscar con Google
rtment of Statistics Online Learning! - Penn State","th":83,"tu":"https://encrypted-tbn0.gstatic.com/images?q\u003dtbn:ANd9GcQuElAJ2v_EaT3kTk6OttMFj8vC8cwGQrbEbwExrMxvAB-IY7aQ1Nkvdoo","tw":210} <span>The empirical rule states that for a normal distribution, nearly all of the data will fall within three standard deviations of the mean. The empirical rule can be broken down into three parts: 68% of data falls within the first standard deviation from the mean. 95% fall within two standard deviations.Nov 1, 2013 Empirical Rule: What is it? - Statistics How To www.statisticshowto.com/empirical-rule-2/ Feedback About this result People also ask What is the empirical rul







The empirical rule can be broken down into three parts: 68% of data falls within the first standard deviation from the mean. 95% fall within two standard deviations.

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The empirical rule states that for a normal distribution, nearly all of the data will fall within three standard deviations of the mean. The empirical rule can be broken down into three parts: 68% of data falls within the first standard deviation from the mean. 95% fall within two standard deviations.

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the+empirical+rule - Buscar con Google
rtment of Statistics Online Learning! - Penn State","th":83,"tu":"https://encrypted-tbn0.gstatic.com/images?q\u003dtbn:ANd9GcQuElAJ2v_EaT3kTk6OttMFj8vC8cwGQrbEbwExrMxvAB-IY7aQ1Nkvdoo","tw":210} <span>The empirical rule states that for a normal distribution, nearly all of the data will fall within three standard deviations of the mean. The empirical rule can be broken down into three parts: 68% of data falls within the first standard deviation from the mean. 95% fall within two standard deviations.Nov 1, 2013 Empirical Rule: What is it? - Statistics How To www.statisticshowto.com/empirical-rule-2/ Feedback About this result People also ask What is the empirical rul




Flashcard 1652329155852

Question
[...] comparisons of quarterly EPS are with the immediate prior quarter.
Answer
Sequential


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Sequential comparisons of quarterly EPS are with the immediate prior quarter. A sequential comparison stands in contrast to a comparison with the same quarter one year ago (another frequent type o







Flashcard 1652332039436

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#reading-9-probability-concepts
Question

the [...] between two random variables is the probability-weighted average of the cross-products of each random variable’s deviation from its own expected value.

Answer
covariance


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the covariance between two random variables is the probability-weighted average of the cross-products of each random variable’s deviation from its own expected value.







Flashcard 1652333612300

Tags
#reading-9-probability-concepts
Question

\(n!\over n1!n2!…nk!\) is called [...]

Answer
Multinomial Formula


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Multinomial Formula (General Formula for Labeling Problems). The number of ways that n objects can be labeled with kdifferent labels, with n 1 of the first type, n 2 of the second type, and so on, with n







Flashcard 1652335971596

Tags
#reading-9-probability-concepts
Question

General Formula for Labeling Problems

Answer
Multinomial Formula


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Multinomial Formula (General Formula for Labeling Problems). The number of ways that n objects can be labeled with kdifferent labels, with n 1 of the first type, n 2 of the second type, and so on, with n







Flashcard 1652339903756

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#reading-9-probability-concepts
Question

The number of ways that n objects can be labeled with k different labels, with n1 of the first type, n2 of the second type, and so on, with n1 + n2 + … + nk = n,

Answer

\(n!\over n1!n2!…nk!\)


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Multinomial Formula (General Formula for Labeling Problems). The number of ways that n objects can be labeled with kdifferent labels, with n 1 of the first type, n 2 of the second type, and so on, with n 1 + n 2 + … + n k = n, is given by \(n!\over n1!n2!…nk!\)







Flashcard 1652342263052

Tags
#reading-9-probability-concepts
Question
The combination formula is also called the [...]
Answer
binomial formula


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>A combination is a listing in which the order of listing does not matter. This describes the number of ways that we can choose r objects from a total of n objects, where the order in which the r objects is listed does not matter (The combination formula, or the binomial formula): <body><html>

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Subject 10. Principles of Counting
nlike the multiplication rule, factorial involves only a single group. It involves arranging items within a group, and the order of the arrangement does matter. The arrangement of ABCDE is different from the arrangement of ACBDE. <span>A combination is a listing in which the order of listing does not matter. This describes the number of ways that we can choose r objects from a total of n objects, where the order in which the r objects is listed does not matter (The combination formula, or the binomial formula): For example, if you select two of the ten stocks you are analyzing, how many ways can you select the stocks? 10! / [(10 - 2)! x 2!] = 45. &







Flashcard 1652344622348

Tags
#has-images #reading-9-probability-concepts
Question
The combination formula, or the binomial formula):
Answer


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A combination is a listing in which the order of listing does not matter. This describes the number of ways that we can choose r objects from a total of n objects, where the order in which the r objects is listed does not matter (The combination formula, or the binomial formula):

Original toplevel document

Subject 10. Principles of Counting
nlike the multiplication rule, factorial involves only a single group. It involves arranging items within a group, and the order of the arrangement does matter. The arrangement of ABCDE is different from the arrangement of ACBDE. <span>A combination is a listing in which the order of listing does not matter. This describes the number of ways that we can choose r objects from a total of n objects, where the order in which the r objects is listed does not matter (The combination formula, or the binomial formula): For example, if you select two of the ten stocks you are analyzing, how many ways can you select the stocks? 10! / [(10 - 2)! x 2!] = 45. &







Flashcard 1652346981644

Tags
#reading-9-probability-concepts
Question
A [...] describes the number of ways that we can choose r objects from a total of n objects, where the order in which the r objects is listed does not matter
Answer
combination


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A combination is a listing in which the order of listing does not matter. This describes the number of ways that we can choose r objects from a total of n objects, where the order in which the r objects is listed does not matter (The combination formula, or the binomial formula):

Original toplevel document

Subject 10. Principles of Counting
nlike the multiplication rule, factorial involves only a single group. It involves arranging items within a group, and the order of the arrangement does matter. The arrangement of ABCDE is different from the arrangement of ACBDE. <span>A combination is a listing in which the order of listing does not matter. This describes the number of ways that we can choose r objects from a total of n objects, where the order in which the r objects is listed does not matter (The combination formula, or the binomial formula): For example, if you select two of the ten stocks you are analyzing, how many ways can you select the stocks? 10! / [(10 - 2)! x 2!] = 45. &







#reading-9-probability-concepts
the sample mean presents a central value for a particular set of observations as an equally weighted average of those observations.

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Expected value
tion mean, discussed in the reading on statistical concepts and market returns). We should distinguish expected value from the concepts of historical or sample mean. The sample mean also summarizes in a single number a central value. However, <span>the sample mean presents a central value for a particular set of observations as an equally weighted average of those observations. To summarize, the contrast is forecast versus historical, or population versus sample.<span><body><html>




#reading-9-probability-concepts
To summarize, the contrast is forecast versus historical, or population versus sample.

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Expected value
pts of historical or sample mean. The sample mean also summarizes in a single number a central value. However, the sample mean presents a central value for a particular set of observations as an equally weighted average of those observations. <span>To summarize, the contrast is forecast versus historical, or population versus sample.<span><body><html>




Flashcard 1652353535244

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#reading-9-probability-concepts
Question
Looking at the dispersion or variance of portfolio return, we see that the way individual security returns move together or [...] is important.
Answer
covary


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To analyze a portfolio’s expected return and variance of return, we must understand these quantities are a function of characteristics of the individual securities’ returns. Looking at the dispersion or variance of portfolio return, we see that the way individual security returns move together or covary is important. To understand the significance of these movements, we need to explore some new concepts, covariance and correlation.







Flashcard 1652357205260

Tags
#reading-8-statistical-concepts-and-market-returns #skewness
Question

In calculations of variance we do not know whether large deviations are likely to be positive or negative, hence the [...] in return distributions.

Answer
degree of symmetry


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In calculations of variance we do not know whether large deviations are likely to be positive or negative, hence the degree of symmetry in return distributions.







Flashcard 1652360350988

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#reading-9-probability-concepts
Question
The word marginal in economics roughly means [...]
Answer
incremental


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l>In analyses of probabilities presented in tables, unconditional probabilities usually appear at the ends or margins of the table, hence the term marginal probability. Because of possible confusion with the way marginal is used in economics (roughly meaning incremental), we use the term unconditional probability throughout this discussion.<html>







Flashcard 1652366642444

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#reading-9-probability-concepts
Question
If we know both the set of all the distinct possible outcomes of a random variable and the assignment of probabilities to those outcomes then we have the [...] of the random variable.
Answer
the probability distribution


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If we know both the set of all the distinct possible outcomes of a random variable and the assignment of probabilities to those outcomes—the probability distribution of the random variable—we have a complete description of the random variable, and we can assign a probability to any event that we might describe







#reading-9-probability-concepts
If we have the probability distribution of a random variable, we can assign a probability to any event that we might describe.

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If we know both the set of all the distinct possible outcomes of a random variable and the assignment of probabilities to those outcomes—the probability distribution of the random variable—we have a complete description of the random variable, and we can assign a probability to any event that we might describe




Multinominal formula
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A mutual fund guide ranked 18 bond mutual funds by total returns for the year 2014. The guide also assigned each fund one of five risk labels: high risk (four funds), above-average risk (four funds), average risk (three funds), below-average risk (four funds), and low risk (three funds); as 4 + 4 + 3 + 4 + 3 = 18, all the funds are accounted for. How many different ways can we take 18 mutual funds and label 4 of them high risk, 4 above-average risk, 3 average risk, 4 below-average risk, and 3 low risk, so that each fund is labeled?

The answer is close to 13 billion. We can label any of 18 funds high risk (the first slot), then any of 17 remaining funds, then any of 16 remaining funds, then any of 15 remaining funds (now we have 4 funds in the high risk group); then we can label any of 14 remaining funds above-average risk, then any of 13 remaining funds, and so forth. There are 18! possible sequences. However, order of assignment within a category does not matter. For example, whether a fund occupies the first or third slot of the four funds labeled high risk, the fund has the same label (high risk). Thus there are 4! ways to assign a given group of four funds to the four high risk slots. Making the same argument for the other categories, in total there are (4!)(4!)(3!)(4!)(3!) equivalent sequences. To eliminate such redundancies from the 18! total, we divide 18! by (4!)(4!)(3!)(4!)(3!). We have 18!/(4!)(4!)(3!)(4!)(3!) = 18!/(24)(24)(6)(24)(6) = 12,864,852,000. This procedure generalizes as follows.

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Flashcard 1652374506764

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Question
The multinomial formula with [...] is especially important.
Answer
two different labels (k = 2)


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Multinominal formula
A mutual fund guide ranked 18 bond mutual funds by total returns for the year 2014. The guide also assigned each fund one of five risk labels: high risk (four funds), above-average risk (







Flashcard 1652376866060

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Question
The multinomial formula with two different labels (k = 2) is called the [...]
Answer
combination formula or binominal formula


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Multinominal formula
A mutual fund guide ranked 18 bond mutual funds by total returns for the year 2014. The guide also assigned each fund one of five risk labels: high risk (four funds), above-average risk (four funds), average risk (t







Flashcard 1652379487500

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Question
A listing in which the order of the listed items does not matter.
Answer
Combination


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Multinominal formula
A mutual fund guide ranked 18 bond mutual funds by total returns for the year 2014. The guide also assigned each fund one of five risk labels: high risk (four funds), above-average risk (







Binominal formula
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Using the notation in the formula below, the number of objects with the first label is r = n1 and the number with the second label is nr = n2 (there are just two categories, so n1 + n2 = n). Here is the formula:

  • Combination Formula (Binomial Formula). The number of ways that we can choose r objects from a total of n objects, when the order in which the r objects are listed does not matter, is

    \(_nC_r=(\frac{n}{r})= \frac{n!}{(n-r)!r!}\)

    Here nCr and (nr) are shorthand notations for n!/(nr)!r! (read: n choose r, or n combination r).

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Binominal formula
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If we label the r objects as belongs to the group and the remaining objects as does not belong to the group, whatever the group of interest, the combination formula tells us how many ways we can select a group of size r. We can illustrate this formula with the binomial option pricing model. This model describes the movement of the underlying asset as a series of moves, price up (U) or price down (D). For example, two sequences of five moves containing three up moves, such as UUUDD and UDUUD, result in the same final stock price. At least for an option with a payoff dependent on final stock price, the number but not the order of up moves in a sequence matters. How many sequences of five moves belong to the group with three up moves? The answer is 10, calculated using the combination formula (“5 choose 3”):

5C3 = 5! /(5−3)!3! =(5)(4)(3)(2)(1)/(2)(1)(3)(2)(1) = 120 / 12 = 10 ways

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Counting
#reading-9-probability-concepts

Answering the following questions may help you apply the counting methods we have presented in this section.

  1. Does the task that I want to measure have a finite number of possible outcomes? If the answer is yes, you may be able to use a tool in this section, and you can go to the second question. If the answer is no, the number of outcomes is infinite, and the tools in this section do not apply.

  2. Do I want to assign every member of a group of size n to one of n slots (or tasks)? If the answer is yes, use n factorial. If the answer is no, go to the third question.

  3. Do I want to count the number of ways to apply one of three or more labels to each member of a group? If the answer is yes, use the multinomial formula. If the answer is no, go to the fourth question.

  4. Do I want to count the number of ways that I can choose r objects from a total of n, when the order in which I list the r objects does not matter (can I give the r objects a label)? If the answer to these questions is yes, the combination formula applies. If the answer is no, go to the fifth question.

  5. Do I want to count the number of ways I can choose r objects from a total of n, when the order in which I list the r objects is important? If the answer is yes, the permutation formula applies. If the answer is no, go to question 6.

  6. Can the multiplication rule of counting be used? If it cannot, you may have to count the possibilities one by one, or use more advanced techniques than those presented here.23

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Flashcard 1652389448972

Tags
#reading-9-probability-concepts
Question
Do I want to assign every member of a group of size n to one of n slots (or tasks)? If the answer is yes, use [...]
Answer
n factorial.


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Counting
ossible outcomes? If the answer is yes, you may be able to use a tool in this section, and you can go to the second question. If the answer is no, the number of outcomes is infinite, and the tools in this section do not apply. <span>Do I want to assign every member of a group of size n to one of n slots (or tasks)? If the answer is yes, use n factorial. If the answer is no, go to the third question. Do I want to count the number of ways to apply one of three or more labels to each member of a group? If the answer is yes







#reading-9-probability-concepts
Does the task that I want to measure have a finite number of possible outcomes? If the answer is yes, you may be able to use a tool in this section, and you can go to the second question. If the answer is no, the number of outcomes is infinite, and the tools in this section do not apply.

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Counting
Answering the following questions may help you apply the counting methods we have presented in this section. Does the task that I want to measure have a finite number of possible outcomes? If the answer is yes, you may be able to use a tool in this section, and you can go to the second question. If the answer is no, the number of outcomes is infinite, and the tools in this section do not apply. Do I want to assign every member of a group of size n to one of n slots (or tasks)? If the answer is yes, use n factorial. If the answer is no, go to the third question.




Flashcard 1652393381132

Tags
#reading-9-probability-concepts
Question
Do I want to count the number of ways to apply one of three or more labels to each member of a group? If the answer is yes, use the [...]
Answer
multinomial formula.


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Counting
the tools in this section do not apply. Do I want to assign every member of a group of size n to one of n slots (or tasks)? If the answer is yes, use n factorial. If the answer is no, go to the third question. <span>Do I want to count the number of ways to apply one of three or more labels to each member of a group? If the answer is yes, use the multinomial formula. If the answer is no, go to the fourth question. Do I want to count the number of ways that I can choose r objects from a total of n, when the order in which I list the r







Flashcard 1652396002572

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#reading-9-probability-concepts
Question
Do I want to count the number of ways that I can choose r objects from a total of n, when the order in which I list the r objects does not matter? If the answer to these questions is yes, use the [...]
Answer
combination formula


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Counting
question. Do I want to count the number of ways to apply one of three or more labels to each member of a group? If the answer is yes, use the multinomial formula. If the answer is no, go to the fourth question. <span>Do I want to count the number of ways that I can choose r objects from a total of n, when the order in which I list the r objects does not matter (can I give the r objects a label)? If the answer to these questions is yes, the combination formula applies. If the answer is no, go to the fifth question. Do I want to count the number of ways I can choose r objects from a total of n, when the order in which I list the r objec







Flashcard 1652398361868

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#reading-9-probability-concepts
Question
Do I want to count the number of ways I can choose r objects from a total of n, when the order in which I list the r objects is important? If the answer is yes use the [...]
Answer
permutation formula applies.


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Counting
al of n, when the order in which I list the r objects does not matter (can I give the r objects a label)? If the answer to these questions is yes, the combination formula applies. If the answer is no, go to the fifth question. <span>Do I want to count the number of ways I can choose r objects from a total of n, when the order in which I list the r objects is important? If the answer is yes, the permutation formula applies. If the answer is no, go to question 6. Can the multiplication rule of counting be used? If it cannot, you may have to count the possibilities one by one, or use more adv







#reading-9-probability-concepts
Can the multiplication rule of counting be used? If it cannot, you may have to count the possibilities one by one, or use more advanced techniques than those presented here

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Counting
ant to count the number of ways I can choose r objects from a total of n, when the order in which I list the r objects is important? If the answer is yes, the permutation formula applies. If the answer is no, go to question 6. <span>Can the multiplication rule of counting be used? If it cannot, you may have to count the possibilities one by one, or use more advanced techniques than those presented here.23 <span><body><html>




Summary
#reading-9-probability-concepts

In this reading, we have discussed the essential concepts and tools of probability. We have applied probability, expected value, and variance to a range of investment problems.

  • A random variable is a quantity whose outcome is uncertain.

  • Probability is a number between 0 and 1 that describes the chance that a stated event will occur.

  • An event is a specified set of outcomes of a random variable.

  • Mutually exclusive events can occur only one at a time. Exhaustive events cover or contain all possible outcomes.

  • The two defining properties of a probability are, first, that 0 ≤ P(E) ≤ 1 (where P(E) denotes the probability of an event E), and second, that the sum of the probabilities of any set of mutually exclusive and exhaustive events equals 1.

  • A probability estimated from data as a relative frequency of occurrence is an empirical probability. A probability drawing on personal or subjective judgment is a subjective probability. A probability obtained based on logical analysis is an a priori probability.

  • A probability of an event E, P(E), can be stated as odds for E = P(E)/[1 − P(E)] or odds against E = [1 − P(E)]/P(E).

  • Probabilities that are inconsistent create profit opportunities, according to the Dutch Book Theorem.

  • A probability of an event not conditioned on another event is an unconditional probability. The unconditional probability of an event A is denoted P(A). Unconditional probabilities are also called marginal probabilities.

  • A probability of an event given (conditioned on) another event is a conditional probability. The probability of an event A given an event B is denoted P(A | B).

  • The probability of both A and B occurring is the joint probability of A and B, denoted P(AB).

  • P(A | B) = P(AB)/P(B), P(B) ≠ 0.

  • The multiplication rule for probabilities is P(AB) = P(A | B)P(B).

  • The probability that A or B occurs, or both occur, is denoted by P(A or B).

  • The addition rule for probabilities is P(A or B) = P(A) + P(B) − P(AB).

  • When events are independent, the occurrence of one event does not affect the probability of occurrence of the other event. Otherwise, the events are dependent.

  • The multiplication rule for independent events states that if A and B are independent events, P(AB) = P(A)P(B). The rule generalizes in similar fashion to more than two events.

  • According to the total probability rule, if S1, S2, …, Sn are mutually exclusive and exhaustive scenarios or events, then P(A) = P(A | S1)P(S1) + P(A | S2)P(S2) + … + P(A | Sn)P(Sn).

  • The expected value of a random variable is a probability-weighted average of the possible outcomes of the random variable. For a random variable X, the expected value of X is denoted E(X).

  • The total probability rule for expected value states that E(X) = E(X | S1)P(S1) + E(X | S2)P(S2) + … + E(X

...

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#reading-9-probability-concepts
A random variable is a quantity whose outcome is uncertain.

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Summary
In this reading, we have discussed the essential concepts and tools of probability. We have applied probability, expected value, and variance to a range of investment problems. A random variable is a quantity whose outcome is uncertain. Probability is a number between 0 and 1 that describes the chance that a stated event will occur. An event is a specified set of outcomes of a random var




#reading-9-probability-concepts
Probability is a number between 0 and 1 that describes the chance that a stated event will occur.

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Summary
sed the essential concepts and tools of probability. We have applied probability, expected value, and variance to a range of investment problems. A random variable is a quantity whose outcome is uncertain. <span>Probability is a number between 0 and 1 that describes the chance that a stated event will occur. An event is a specified set of outcomes of a random variable. Mutually exclusive events can occur only one at a time. Exhaustive events cover or contain




#reading-9-probability-concepts
An event is a specified set of outcomes of a random variable.

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Summary
a range of investment problems. A random variable is a quantity whose outcome is uncertain. Probability is a number between 0 and 1 that describes the chance that a stated event will occur. <span>An event is a specified set of outcomes of a random variable. Mutually exclusive events can occur only one at a time. Exhaustive events cover or contain all possible outcomes. The two defining properties of a probab




#reading-9-probability-concepts
Mutually exclusive events can occur only one at a time. Exhaustive events cover or contain all possible outcomes.

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Summary
antity whose outcome is uncertain. Probability is a number between 0 and 1 that describes the chance that a stated event will occur. An event is a specified set of outcomes of a random variable. <span>Mutually exclusive events can occur only one at a time. Exhaustive events cover or contain all possible outcomes. The two defining properties of a probability are, first, that 0 ≤ P(E) ≤ 1 (where P(E) denotes the probability of an event E), and second, that the sum of the probabilit




#reading-9-probability-concepts
The two defining properties of a probability are, first, that 0 ≤ P(E) ≤ 1 (where P(E) denotes the probability of an event E), and second, that the sum of the probabilities of any set of mutually exclusive and exhaustive events equals 1.

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Summary
event will occur. An event is a specified set of outcomes of a random variable. Mutually exclusive events can occur only one at a time. Exhaustive events cover or contain all possible outcomes. <span>The two defining properties of a probability are, first, that 0 ≤ P(E) ≤ 1 (where P(E) denotes the probability of an event E), and second, that the sum of the probabilities of any set of mutually exclusive and exhaustive events equals 1. A probability estimated from data as a relative frequency of occurrence is an empirical probability. A probability drawing on personal or subjective judgment is a subjec




#reading-9-probability-concepts
A probability estimated from data as a relative frequency of occurrence is an empirical probability. A probability drawing on personal or subjective judgment is a subjective probability. A probability obtained based on logical analysis is an a priori probability.

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Summary
ining properties of a probability are, first, that 0 ≤ P(E) ≤ 1 (where P(E) denotes the probability of an event E), and second, that the sum of the probabilities of any set of mutually exclusive and exhaustive events equals 1. <span>A probability estimated from data as a relative frequency of occurrence is an empirical probability. A probability drawing on personal or subjective judgment is a subjective probability. A probability obtained based on logical analysis is an a priori probability. A probability of an event E, P(E), can be stated as odds for E = P(E)/[1 − P(E)] or odds against E = [1 − P(E)]/P(E). Probabilities that are inconsistent




#reading-9-probability-concepts
A probability of an event E, P(E), can be stated as odds for E = P(E)/[1 − P(E)] or odds against E = [1 − P(E)]/P(E).

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Summary
a relative frequency of occurrence is an empirical probability. A probability drawing on personal or subjective judgment is a subjective probability. A probability obtained based on logical analysis is an a priori probability. <span>A probability of an event E, P(E), can be stated as odds for E = P(E)/[1 − P(E)] or odds against E = [1 − P(E)]/P(E). Probabilities that are inconsistent create profit opportunities, according to the Dutch Book Theorem. A probability of an event not conditioned on anothe




#reading-9-probability-concepts
Probabilities that are inconsistent create profit opportunities, according to the Dutch Book Theorem.

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Summary
ve probability. A probability obtained based on logical analysis is an a priori probability. A probability of an event E, P(E), can be stated as odds for E = P(E)/[1 − P(E)] or odds against E = [1 − P(E)]/P(E). <span>Probabilities that are inconsistent create profit opportunities, according to the Dutch Book Theorem. A probability of an event not conditioned on another event is an unconditional probability. The unconditional probability of an event A is denoted P(A). Unconditional pr




#reading-9-probability-concepts
A probability of an event not conditioned on another event is an unconditional probability. The unconditional probability of an event A is denoted P(A). Unconditional probabilities are also called marginal probabilities.

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Summary
lity of an event E, P(E), can be stated as odds for E = P(E)/[1 − P(E)] or odds against E = [1 − P(E)]/P(E). Probabilities that are inconsistent create profit opportunities, according to the Dutch Book Theorem. <span>A probability of an event not conditioned on another event is an unconditional probability. The unconditional probability of an event A is denoted P(A). Unconditional probabilities are also called marginal probabilities. A probability of an event given (conditioned on) another event is a conditional probability. The probability of an event A given an event B is denoted P(A | B).




#reading-9-probability-concepts
A probability of an event given (conditioned on) another event is a conditional probability. The probability of an event A given an event B is denoted P(A | B).

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Summary
13; A probability of an event not conditioned on another event is an unconditional probability. The unconditional probability of an event A is denoted P(A). Unconditional probabilities are also called marginal probabilities. <span>A probability of an event given (conditioned on) another event is a conditional probability. The probability of an event A given an event B is denoted P(A | B). The probability of both A and B occurring is the joint probability of A and B, denoted P(AB). P(A | B) = P(AB)/P(B), P(B) ≠ 0. The multip




#reading-9-probability-concepts
The probability of both A and B occurring is the joint probability of A and B, denoted P(AB).

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Summary
abilities are also called marginal probabilities. A probability of an event given (conditioned on) another event is a conditional probability. The probability of an event A given an event B is denoted P(A | B). <span>The probability of both A and B occurring is the joint probability of A and B, denoted P(AB). P(A | B) = P(AB)/P(B), P(B) ≠ 0. The multiplication rule for probabilities is P(AB) = P(A | B)P(B). The probability that A or B occurs, o




#reading-9-probability-concepts
P(A | B) = P(AB)/P(B), P(B) ≠ 0.

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Summary
on) another event is a conditional probability. The probability of an event A given an event B is denoted P(A | B). The probability of both A and B occurring is the joint probability of A and B, denoted P(AB). <span>P(A | B) = P(AB)/P(B), P(B) ≠ 0. The multiplication rule for probabilities is P(AB) = P(A | B)P(B). The probability that A or B occurs, or both occur, is denoted by P(A or B). &#




#reading-9-probability-concepts
The multiplication rule for probabilities is P(AB) = P(A | B)P(B).

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Summary
The probability of an event A given an event B is denoted P(A | B). The probability of both A and B occurring is the joint probability of A and B, denoted P(AB). P(A | B) = P(AB)/P(B), P(B) ≠ 0. <span>The multiplication rule for probabilities is P(AB) = P(A | B)P(B). The probability that A or B occurs, or both occur, is denoted by P(A or B). The addition rule for probabilities is P(A or B) = P(A) + P(B) − P(AB).




#reading-9-probability-concepts
The probability that A or B occurs, or both occur, is denoted by P(A or B).

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Summary
The probability of both A and B occurring is the joint probability of A and B, denoted P(AB). P(A | B) = P(AB)/P(B), P(B) ≠ 0. The multiplication rule for probabilities is P(AB) = P(A | B)P(B). <span>The probability that A or B occurs, or both occur, is denoted by P(A or B). The addition rule for probabilities is P(A or B) = P(A) + P(B) − P(AB). When events are independent, the occurrence of one event does not affect the prob




#reading-9-probability-concepts
The addition rule for probabilities is P(A or B) = P(A) + P(B) − P(AB).

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Summary
). P(A | B) = P(AB)/P(B), P(B) ≠ 0. The multiplication rule for probabilities is P(AB) = P(A | B)P(B). The probability that A or B occurs, or both occur, is denoted by P(A or B). <span>The addition rule for probabilities is P(A or B) = P(A) + P(B) − P(AB). When events are independent, the occurrence of one event does not affect the probability of occurrence of the other event. Otherwise, the events are dependent. &




#reading-9-probability-concepts
When events are independent, the occurrence of one event does not affect the probability of occurrence of the other event. Otherwise, the events are dependent.

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Summary
ule for probabilities is P(AB) = P(A | B)P(B). The probability that A or B occurs, or both occur, is denoted by P(A or B). The addition rule for probabilities is P(A or B) = P(A) + P(B) − P(AB). <span>When events are independent, the occurrence of one event does not affect the probability of occurrence of the other event. Otherwise, the events are dependent. The multiplication rule for independent events states that if A and B are independent events, P(AB) = P(A)P(B). The rule generalizes in similar fashion to more than two




#reading-9-probability-concepts
The multiplication rule for independent events states that if A and B are independent events, P(AB) = P(A)P(B). The rule generalizes in similar fashion to more than two events.

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Summary
probabilities is P(A or B) = P(A) + P(B) − P(AB). When events are independent, the occurrence of one event does not affect the probability of occurrence of the other event. Otherwise, the events are dependent. <span>The multiplication rule for independent events states that if A and B are independent events, P(AB) = P(A)P(B). The rule generalizes in similar fashion to more than two events. According to the total probability rule, if S 1 , S 2 , …, S n are mutually exclusive and exhaustive scenarios or events, then P(A) = P(A | S 1 )P(S 1 ) + P(A | S 2 )P(




#reading-9-probability-concepts
According to the total probability rule, if S1, S2, …, Sn are mutually exclusive and exhaustive scenarios or events, then P(A) = P(A | S1)P(S1) + P(A | S2)P(S2) + … + P(A | Sn)P(Sn).

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Summary
erwise, the events are dependent. The multiplication rule for independent events states that if A and B are independent events, P(AB) = P(A)P(B). The rule generalizes in similar fashion to more than two events. <span>According to the total probability rule, if S 1 , S 2 , …, S n are mutually exclusive and exhaustive scenarios or events, then P(A) = P(A | S 1 )P(S 1 ) + P(A | S 2 )P(S 2 ) + … + P(A | S n )P(S n ). The expected value of a random variable is a probability-weighted average of the possible outcomes of the random variable. For a random variable X, the expected value of




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The expected value of a random variable is a probability-weighted average of the possible outcomes of the random variable. For a random variable X, the expected value of X is denoted E(X).

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o events. According to the total probability rule, if S 1 , S 2 , …, S n are mutually exclusive and exhaustive scenarios or events, then P(A) = P(A | S 1 )P(S 1 ) + P(A | S 2 )P(S 2 ) + … + P(A | S n )P(S n ). <span>The expected value of a random variable is a probability-weighted average of the possible outcomes of the random variable. For a random variable X, the expected value of X is denoted E(X). The total probability rule for expected value states that E(X) = E(X | S 1 )P(S 1 ) + E(X | S 2 )P(S 2 ) + … + E(X | S n )P(S n ), where S 1 , S 2 , …, S n are mutually




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The total probability rule for expected value states that E(X) = E(X | S1)P(S1) + E(X | S2)P(S2) + … + E(X | Sn)P(Sn), where S1, S2, …, Sn are mutually exclusive and exhaustive scenarios or events.

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+ P(A | S n )P(S n ). The expected value of a random variable is a probability-weighted average of the possible outcomes of the random variable. For a random variable X, the expected value of X is denoted E(X). <span>The total probability rule for expected value states that E(X) = E(X | S 1 )P(S 1 ) + E(X | S 2 )P(S 2 ) + … + E(X | S n )P(S n ), where S 1 , S 2 , …, S n are mutually exclusive and exhaustive scenarios or events. The variance of a random variable is the expected value (the probability-weighted average) of squared deviations from the random variable’s expected value E(X): σ 2 (X)




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The variance of a random variable is the expected value (the probability-weighted average) of squared deviations from the random variable’s expected value E(X): σ2(X) = E{[XE(X)]2}, where σ2(X) stands for the variance of X.

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The total probability rule for expected value states that E(X) = E(X | S 1 )P(S 1 ) + E(X | S 2 )P(S 2 ) + … + E(X | S n )P(S n ), where S 1 , S 2 , …, S n are mutually exclusive and exhaustive scenarios or events. <span>The variance of a random variable is the expected value (the probability-weighted average) of squared deviations from the random variable’s expected value E(X): σ 2 (X) = E{[X − E(X)] 2 }, where σ 2 (X) stands for the variance of X. Variance is a measure of dispersion about the mean. Increasing variance indicates increasing dispersion. Variance is measured in squared units of the original variable.




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Variance is a measure of dispersion about the mean. Increasing variance indicates increasing dispersion. Variance is measured in squared units of the original variable.

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riance of a random variable is the expected value (the probability-weighted average) of squared deviations from the random variable’s expected value E(X): σ 2 (X) = E{[X − E(X)] 2 }, where σ 2 (X) stands for the variance of X. <span>Variance is a measure of dispersion about the mean. Increasing variance indicates increasing dispersion. Variance is measured in squared units of the original variable. Standard deviation is the positive square root of variance. Standard deviation measures dispersion (as does variance), but it is measured in the same units as the variab




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Standard deviation is the positive square root of variance. Standard deviation measures dispersion (as does variance), but it is measured in the same units as the variable.

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ere σ 2 (X) stands for the variance of X. Variance is a measure of dispersion about the mean. Increasing variance indicates increasing dispersion. Variance is measured in squared units of the original variable. <span>Standard deviation is the positive square root of variance. Standard deviation measures dispersion (as does variance), but it is measured in the same units as the variable. Covariance is a measure of the co-movement between random variables. The covariance between two random variables R i and R j is the expected value of t




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Covariance is a measure of the co-movement between random variables

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uared units of the original variable. Standard deviation is the positive square root of variance. Standard deviation measures dispersion (as does variance), but it is measured in the same units as the variable. <span>Covariance is a measure of the co-movement between random variables. The covariance between two random variables R i and R j is the expected value of the cross-product of the deviations of the two random variables from their respective




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The covariance between two random variables Ri and Rj is the expected value of the cross-product of the deviations of the two random variables from their respective means: Cov(Ri,Rj) = E{[RiE(Ri)][RjE(Rj)]}. The covariance of a random variable with itself is its own variance.

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ive square root of variance. Standard deviation measures dispersion (as does variance), but it is measured in the same units as the variable. Covariance is a measure of the co-movement between random variables. <span>The covariance between two random variables R i and R j is the expected value of the cross-product of the deviations of the two random variables from their respective means: Cov(R i ,R j ) = E{[R i − E(R i )][R j − E(R j )]}. The covariance of a random variable with itself is its own variance. Correlation is a number between −1 and +1 that measures the co-movement (linear association) between two random variables: ρ(R i ,R j ) = Cov(R i ,R j )/[σ(R i ) σ(R j )




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Correlation is a number between −1 and +1 that measures the co-movement (linear association) between two random variables: ρ(Ri,Rj) = Cov(Ri,Rj)/[σ(Ri) σ(Rj)].

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d value of the cross-product of the deviations of the two random variables from their respective means: Cov(R i ,R j ) = E{[R i − E(R i )][R j − E(R j )]}. The covariance of a random variable with itself is its own variance. <span>Correlation is a number between −1 and +1 that measures the co-movement (linear association) between two random variables: ρ(R i ,R j ) = Cov(R i ,R j )/[σ(R i ) σ(R j )]. To calculate the variance of return on a portfolio of n assets, the inputs needed are the n expected returns on the individual assets, n variances of return on the indiv




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To calculate the variance of return on a portfolio of n assets, the inputs needed are the n expected returns on the individual assets, n variances of return on the individual assets, and n(n − 1)/2 distinct covariances.

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iable with itself is its own variance. Correlation is a number between −1 and +1 that measures the co-movement (linear association) between two random variables: ρ(R i ,R j ) = Cov(R i ,R j )/[σ(R i ) σ(R j )]. <span>To calculate the variance of return on a portfolio of n assets, the inputs needed are the n expected returns on the individual assets, n variances of return on the individual assets, and n(n − 1)/2 distinct covariances. Portfolio variance of return is σ2(Rp)=n∑i=1n∑j=1wiwjCov(Ri,Rj)σ2(Rp)=∑i=1n∑j=1nwiwjCov(Ri,Rj) . The calculation of covariance in a forward-looking sense




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Portfolio variance of return is \(σ^2(Rp) = \displaystyle\sum_{i=1}^{n}\displaystyle\sum_{j=1}^{n}w_iw_jCov (R_i,R_j)\)

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#13; To calculate the variance of return on a portfolio of n assets, the inputs needed are the n expected returns on the individual assets, n variances of return on the individual assets, and n(n − 1)/2 distinct covariances. <span>Portfolio variance of return is σ2(Rp)=n∑i=1n∑j=1wiwjCov(Ri,Rj)σ2(Rp)=∑i=1n∑j=1nwiwjCov(Ri,Rj) . The calculation of covariance in a forward-looking sense requires the specification of a joint probability function, which gives the probability of joint occurrences of




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The calculation of covariance in a forward-looking sense requires the specification of a joint probability function, which gives the probability of joint occurrences of values of the two random variables.

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ns on the individual assets, n variances of return on the individual assets, and n(n − 1)/2 distinct covariances. Portfolio variance of return is σ2(Rp)=n∑i=1n∑j=1wiwjCov(Ri,Rj)σ2(Rp)=∑i=1n∑j=1nwiwjCov(Ri,Rj) . <span>The calculation of covariance in a forward-looking sense requires the specification of a joint probability function, which gives the probability of joint occurrences of values of the two random variables. When two random variables are independent, the joint probability function is the product of the individual probability functions of the random variables.




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When two random variables are independent, the joint probability function is the product of the individual probability functions of the random variables.

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Rj) . The calculation of covariance in a forward-looking sense requires the specification of a joint probability function, which gives the probability of joint occurrences of values of the two random variables. <span>When two random variables are independent, the joint probability function is the product of the individual probability functions of the random variables. Bayes’ formula is a method for updating probabilities based on new information. Bayes’ formula is expressed as follows: Updated probability of event give




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Bayes’ formula is a method for updating probabilities based on new information.

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joint occurrences of values of the two random variables. When two random variables are independent, the joint probability function is the product of the individual probability functions of the random variables. <span>Bayes’ formula is a method for updating probabilities based on new information. Bayes’ formula is expressed as follows: Updated probability of event given the new information = [(Probability of the new information given event)/(Unconditional probabi




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Bayes’ formula is expressed as follows: Updated probability of event given the new information = [(Probability of the new information given event)/(Unconditional probability of the new information)] × Prior probability of event.

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es are independent, the joint probability function is the product of the individual probability functions of the random variables. Bayes’ formula is a method for updating probabilities based on new information. <span>Bayes’ formula is expressed as follows: Updated probability of event given the new information = [(Probability of the new information given event)/(Unconditional probability of the new information)] × Prior probability of event. The multiplication rule of counting says, for example, that if the first step in a process can be done in 10 ways, the second step, given the first, can be done in 5 way




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The multiplication rule of counting says, for example, that if the first step in a process can be done in 10 ways, the second step, given the first, can be done in 5 ways, and the third step, given the first two, can be done in 7 ways, then the steps can be carried out in (10)(5)(7) = 350 ways.

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yes’ formula is expressed as follows: Updated probability of event given the new information = [(Probability of the new information given event)/(Unconditional probability of the new information)] × Prior probability of event. <span>The multiplication rule of counting says, for example, that if the first step in a process can be done in 10 ways, the second step, given the first, can be done in 5 ways, and the third step, given the first two, can be done in 7 ways, then the steps can be carried out in (10)(5)(7) = 350 ways. The number of ways to assign every member of a group of size n to n slots is n! = n (n − 1) (n − 2)(n − 3) … 1. (By convention, 0! = 1.) The number of wa




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The number of ways to assign every member of a group of size n to n slots is n! = n (n − 1) (n − 2)(n − 3) … 1. (By convention, 0! = 1.)

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rst step in a process can be done in 10 ways, the second step, given the first, can be done in 5 ways, and the third step, given the first two, can be done in 7 ways, then the steps can be carried out in (10)(5)(7) = 350 ways. <span>The number of ways to assign every member of a group of size n to n slots is n! = n (n − 1) (n − 2)(n − 3) … 1. (By convention, 0! = 1.) The number of ways that n objects can be labeled with k different labels, with n 1 of the first type, n 2 of the second type, and so on, with n 1 + n 2 + … + n k =




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The number of ways that n objects can be labeled with k different labels, with n1 of the first type, n2 of the second type, and so on, with n1 + n2 + … + nk = n, is given by n!/(n1!n2! … nk!). This expression is the multinomial formula.

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ne in 7 ways, then the steps can be carried out in (10)(5)(7) = 350 ways. The number of ways to assign every member of a group of size n to n slots is n! = n (n − 1) (n − 2)(n − 3) … 1. (By convention, 0! = 1.) <span>The number of ways that n objects can be labeled with k different labels, with n 1 of the first type, n 2 of the second type, and so on, with n 1 + n 2 + … + n k = n, is given by n!/(n 1 !n 2 ! … n k !). This expression is the multinomial formula. A special case of the multinomial formula is the combination formula. The number of ways to choose r objects from a total of n objects, when the order in which the robje




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A special case of the multinomial formula is the combination formula. The number of ways to choose r objects from a total of n objects, when the order in which the robjects are listed does not matter, is

nCr=(nr)=n!(n−r)!r!nCr=(nr)=n!(n−r)!r!

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objects can be labeled with k different labels, with n 1 of the first type, n 2 of the second type, and so on, with n 1 + n 2 + … + n k = n, is given by n!/(n 1 !n 2 ! … n k !). This expression is the multinomial formula. <span>A special case of the multinomial formula is the combination formula. The number of ways to choose r objects from a total of n objects, when the order in which the robjects are listed does not matter, is nCr=(nr)=n!(n−r)!r!nCr=(nr)=n!(n−r)!r! The number of ways to choose r objects from a total of n objects, when the order in which the r objects are listed does matter, is nPr=n!(n




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The number of ways to choose r objects from a total of n objects, when the order in which the r objects are listed does matter, is

nPr=n!(n−r)!nPr=n!(n−r)!

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he combination formula. The number of ways to choose r objects from a total of n objects, when the order in which the robjects are listed does not matter, is nCr=(nr)=n!(n−r)!r!nCr=(nr)=n!(n−r)!r! <span>The number of ways to choose r objects from a total of n objects, when the order in which the r objects are listed does matter, is nPr=n!(n−r)!nPr=n!(n−r)! This expression is the permutation formula. <span><body><html>




Flashcard 1652462587148

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Question
nPr = [...]
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statusnot learnedmeasured difficulty37% [default]last interval [days]               
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Subject 10. Principles of Counting
(The combination formula, or the binomial formula): For example, if you select two of the ten stocks you are analyzing, how many ways can you select the stocks? 10! / [(10 - 2)! x 2!] = 45. <span>An ordered listing is known as a permutation, and the formula that counts the number of permutations is known as the permutation formula. The number of ways that we can choose r objects from a total of n objects, where the order in which the r objects is listed does matter, is: For example, if you select two of the ten stocks you are analyzing and invest $10,000 in one stock and $20,000 in another stock, how many ways ca







Flashcard 1652464946444

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Question
nCr =
Answer


statusnot learnedmeasured difficulty37% [default]last interval [days]               
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Subject 10. Principles of Counting
A combination is a listing in which the order of listing does not matter. This describes the number of ways that we can choose r objects from a total of n objects, where the order in which the r objects is listed does not matter (<span>The combination formula, or the binomial formula): For example, if you select two of the ten stocks you are analyzing, how many ways can you select the stocks? 10! / [(10 - 2)! x 2!] = 45. &