Definition and Examples of Expected Value

Definitions

Suppose that X is a random variable for an experiment, taking values in a subset S of R. If X has a discrete distribution with density function f then the expected value of X is defined by

If X has a continuous distribution with density function f then the expected value of X is defined by

Finally suppose that X has a mixed distribution. Specifically, assume that S is the union of disjoint sets D and C such that the conditional distribution of X given X in D is discrete with density f_D and the conditional distribution of X given X in C is continuous with density f_C . Let p = P(X in D). The expected value of X is

In any case, the expected value of X may not exist because the sum or the integral may not converge. However, unless otherwise noted, whenever we write an expected value, we are assuming that it exists.

The mean is the center of the probability distribution of X. Indeed, if we think of the distribution as a mass distribution, then the mean is the center of mass as defined in physics.

To understand expected value in a probabilistic way, suppose that we create a new, compound experiment by repeating the basic experiment over and over again. This gives a sequence of independent random variables,

X₁, X₂, X₃ ...

each with the same distribution as X. In statistical terms, we are sampling from the distribution of X. The average value, or sample mean, after n runs is

Then the average value will converge to the expected value as n increases. The precise statement of this is the law of large numbers.

Examples

$Mathematical Exercise$ 1. Let I be an indicator random variable (that is, a variable that takes only the values 0 and 1). Show that

E(I) = P(I = 1)

$Mathematical Exercise$ 2. Suppose that X is uniformly distributed on a finite subset S of R. Show that E(X) is the arithmetic average of the numbers in S.

$Mathematical Exercise$ 3. Suppose that X is uniformly distributed on an interval (a, b) of R. Show that the mean is the midpoint of the interval:

E(X) = (a + b) / 2

$Mathematical Exercise$ 4. Find the mean of the random variable X that has probability density function f given by

f(x) = x² / 3 for -1 < x < 2

$Mathematical Exercise$ 5. Suppose that X has the power distribution with parameter a > 1, which has density

f(x) = (a - 1)x^-a for x > 1

Show that

$Mathematical Exercise$ 5. Suppose that X has the Cauchy distribution with density function

Show that the expected value of X does not exist.