EXPECTED VALUE AND VARIANCE Its important to compute

EXPECTED VALUE AND VARIANCE • It’s important to compute mean (expected value) and variance of probability distribution. For example, – Recall from our discussion on random variables and random numbers that if we want to generate random numbers, it may be necessary to specify mean and variance (along with the distribution) of the random numbers. – Suppose that you have to decide whether or not to make an investment that has an uncertain return. You may like to know whether the expected return is more than the investment. 1

EXPECTED VALUE • The expected value is obtained as follows: • E(X) is the expected value of the random variable X • xi is the i-th possible value of the random variable X • p(xi) is the probability that the random variable X will assume the value xi 2

EXPECTED VALUE: EXAMPLE Example 1: Hale’s TV productions is considering producing a pilot for a comedy series for a major television network. While the network may reject the pilot and the series, it may also purchase the program for 1 or 2 years. Hale’s payoffs (profits and losses in $1000 s) and probabilities of the events are summarized below: What should the company do? 3

LAWS OF EXPECTED VALUE • The laws of expected value are listed below: • X and Y are random variables • c is a constant • E(X), E(Y), and E(c) are expected values of X, Y and c respectively. 4

LAWS OF EXPECTED VALUE: EXAMPLE Example 2: If it turns out that each payoff value of Hale’s TV is overestimated by $50, 000, what the company should do? 5

LAWS OF EXPECTED VALUE: EXAMPLE Example 3: Tucson Machinery Inc. manufactures Computer Numerical Controlled (CNC) machines. Sales for the CNC machines are expected to be 30, 36, 42, and 33 units in fall, winter, spring and summer respectively. What is the expected annual sales? 6

LAWS OF EXPECTED VALUE: EXAMPLE Example 4: Let X be a random variable with the following probability distribution: Compute 7

VARIANCE • The variance and standard deviation are obtained as follows: • is the mean (expected value) of random variable X • E[(X- )2] is the variance of random variable X, expected value of squared deviations from the mean • xi is the i-th possible value of random variable X • p(xi) is the probability that random variable X will assume the value xi 8

VARIANCE: EXAMPLE Example 5: Let X be a random variable with the following probability distribution: Compute variance. 9

SHORTCUT FORMULA FOR VARIANCE • The shortcut formula for variance and deviation are as follows: • is the mean (expected value) of random variable X • E(X 2) is the expected value of X 2 and is obtained as follows: • xi is the i-th possible value of random variable X • p(xi) is the probability that random variable X will assume the value xi 10

SHORTCUT FORMULA FOR VARIANCE EXAMPLE Example 6: Let X be a random variable with the following probability distribution: Compute variance using the shortcut formula. 11

LAWS OF VARIANCE • The laws of expected value are listed below: • X and Y are random variables • c is a constant • V(X), V(Y), and V(c) are variances of X, Y and c respectively. 12

LAWS OF VARIANCE: EXAMPLE Example 7: Let X be a random variable with the following probability distribution: Compute 13