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{[X − E(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{[Ri − E(Ri)][Rj − E(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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