MATH 367, fall 98, CHAPTER 4

Section 1,2 RANDOM VARIABLES, EXPECTED VALUE, VARIANCE

Definition: A random variable is a function X : Ω → ℝ.

1. Describe the following random variables, that is, describe what values does it take and with what probabilities: a. Toss a coin. The random viable X records whether it was heads or tails. b. Throw a die. The random variable X records the number on the top face. c. Throw a die 10 times. The random variable X records the total number of “6”’s obtained. d. Throw a die till it falls “6” for the first time. The random variable X records the number of

trials necessary. e. Throw a die till it falls “6” for the fourth time. The random variable X records the number of

trials necessary.

Definition: The expected value of a random variable X is its mean value

E(X) = ∑_{x value of X} xP (X = x).

2. Compute the following: a. the expected value of the random variable in problem 1.a. b. the expected value of the random variable in problem 1.b. c. In a certain lottery, 5,000,000 tickets of $1 are sold each week. The following prices are awarded:

5 prices of $100,00 each 20 prices of $25,00 each 45 prices of $5,00 each 100 prices of $1000 each 1950 prices of $100 each 24500 prices of $10 each

Find the expected value of a single ticket and the expected gain of a single ticket.

Definition: The variance of a random variable X is its mean deviation from the mean

Var(X) = E((X − m)^{2}) = E(X^{2}) − [m]^{2} = ∑_{x value of X}[x − m)]^{2}P (X = x),

where m = E(X).

3. Compute the variances of the random variables in problems 1.a, 1.b, and 2.c.

For next class read sections 4.1 and 4.2 and do problems 1–7, 10–16. (Do as many as you need to get used to computing the expected value and the variance. In particular I want problems: 1, 2, 4, 6, 14.