Aug 06

Expected value of multivariate probability distribution (Quant: Stat)

by David Harper, CFA, FRM, CIPM

Learning objective: Define, calculate and interpret the expected value of multivariate probability distributions

The expected value of a multivariate probability is given by:

In Gujarati, this idea is illustrated with the following example:

Over 200 days, a computer store tracks the number of printers and PCs sold each day. For example, for six out of the 200 days, the store sold zero of both; for 12 out of 200 days, the store sold one printer and two PCs

The upper matrix is the frequency distribution; the lower matrix translates the frequencies into probabilities.

Key points for the FRM candidate:

The bivariate distribution (lower matrix) is a class of multivariate distribution; bivariate is just the simple two-variable case, as opposed to three-, four- or more variables
The lower matrix is a joint probability mass function (joint PMF). If the variable is discrete, it’s a PMF; if the variable is continuous, it’s a probability density function (PDF).
The light green totals give marginal or unconditional probabilities. For example, the marginal (unconditional) probability that three printers are sold equals 27% (0.27)
An example of a conditional probability is, “What is the probability that 4 printers are sold given that (conditional on) 4 PCs are sold?” Put mathematically, P(Y=4 | X=4) = .15/.32
An example of a joint probability is ““What is the probability that 4 printers are sold and 4 PCs are sold?” Put mathematically, P(Y=4, X=4) = .15