MATH 367, spg 98, CHAPTER 4 Section 2, Mar 4 PROPERTIES OF EXPECTED VALUE AND VARIANCE

1. The expected value is linear. Let a, b and c be scalars and X and Y random variables. a. E(c) = c. b. E(a X + b) = a E(X) + b. c. E(X + Y) = E(X) + E(Y). d. If a_{1}, …, a_{n} are scalars and X_{1}, …, X_{n} are random variables then

E(a_{1} X_{1} + … + a_{n} X_{n}) = a_{1}E(X_{1}) + … + a_{n}E(X_{n}).

2. The variance is not linearA. Let X be a random variable and a and b scalars. Then

Var(aX + b) = a^{2}Var(X).

3. Independence makes life easier. If X and Y are independent then a. E(X Y) = E(X) E(Y). b. Var(X + Y) =Var(X)+Var(Y).

Definition X_{1}, …, X_{n} are independent random variables if and only if

P (X_{i_{1}} = a_{1} X_{i_{2}} = a_{2} … X_{i_{r}} = a_{r}) = P (X_{i_{1}} = a_{1}) P (X_{i_{2}} = a_{2}) … P (X_{i_{r}} = a_{r})

for any choice of indices 1 ≤ i_{1} < i_{2} < … < i_{r} ≤ n.

4. If X_{1}, …, X_{n} are independent random variables, then

Var(X_{1} + … + X_{n}) =Var(X_{1}) + … +Var(X_{n}).

5. Computing the mean and variance of famous random variables. a. Let X be a B(n, p) then E(X) = np and Var(X) = npq. b. Let Y a geometric random variable with parameter p. Then E(Y) = 1/p. c. Let Y be the number of trials necessary to obtain r successes. The probability of success in any

given trial is p. Then E(Y) = r/p.

For next class do problems 19, 22, 23, 24, 25, 26, 27, 28, 29, 33, 35, 36, 37, 40, 41, 43, 44, 45, 46, 51, 56.