Expected Values

A motivating example Consider a university having 15,000 students and let X=number of courses for which a randomly selected student is registered. The pmf of X is below. Since p(1)=.01, we know that (.01)(15,000)=150 students are registered for one course, and similarly for other numbers of courses. x 1 2 3 4 5 6 7 p(x) .01 .03 .13 .25 .39 .17 .02 # 150 450 1950 3750 5850 2550 300

Example (average number of classes) To obtain the average number of courses per student, we compute the total number of courses taken and divide the total number of students. Since 150/15,000=.01=p(1), 450/15,000 =.03=p(2) etc., the formula for the average value could also be written as

Implication for formula Thus, to compute the population average value, we only need the possible values for X and the probabilities of X taking on those values.

Definition Let X be a discrete rv with the set of possible values D and pmf p(x). The expected value (or mean) of X, denoted by E(X) or , is

Example: Number of computers in use at Cal Poly 1 2 3 4 5 6 p(x) .05 .10 .15 .25 .20

Example: Bernoulli random variable with parameter p

Definition: Expected value of a function of a random variable If the rv X has a set of possible values D and pmf p(x), then the expected value of any function h(X) is computed by

Example A computer store purchases three computers for $500 apiece. The manufacturer agrees to repurchase any unsold computers after a specified period at $200 each. Let X denote the number of computers sold, and supposed that p(0)=.1, p(1)=.2, p(2)=.3, p(3)=.4. Let h(X) denote the profit associated with selling X units. If the computers are sold for $1000, what is the expected profit?

Solution

Expected value of a linear function Proof: Two special cases are and

Example: computer store p(0)=.1, p(1)=.2, p(2)=.3, p(3)=.4

Expected value as a measure of central location The expected value of X is a measure of where the probability mass is centered. Two distributions can have the same expected value, but the probability mass can be spread differently. Variance gives a measure of spread about the mean.

Definition of variance Let X have pmf p(x) and expected value . The variance of X, , is The standard deviation of X is

Shortcut formula for computing variance It is easier to use the following formula to compute the variance Proof:

Variance of a linear function Proof:

Example: computer store p(0)=.1, p(1)=.2, p(2)=.3, p(3)=.4