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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 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 | Sn)P(Sn), where S1, S2, …, Sn 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): σ2(X) = E{[XE(X)]2}, where σ2(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.

• 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 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.

• 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)].

• 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 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.

• 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 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 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 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.

• 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−r)!nPr=n!(n−r)!

This expression is the permutation formula.

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