CS 237, Fall 2007

Date Due: Thursday, November 8

Reading: Chapter 4, pages 61-72 of the text.

Problems:

1. A card is drawn from a regular 52-card deck at random. Random Variable X is the face value of that card (Ace = 1, Jack = 11, Queen = 12, and King = 13).

a) What is the expected value of X? What is the variance?

b) What does the Markov Inequality give you for this random variable X ?

In particular, what is the probability that a card has face value 11 or higher ?

c) What does Chebyshev's Inequality give you for this same problem ?

d) Which of the 2 inequalities gives you the better bound here ?

Are either of these bounds close to optimal ? Why or why not ?

2. Page 57, problem 3.3

3. Use the Chebyshev inequality to bound the following probability.

A company manufactures light bulbs with a lifetime in months that is binomially distributed with mean 3 and variance 1. You buy a number of these bulbs with the intention of replacing them as soon as they burn out. The lifetimes of the bulbs are independent.

What is the smallest number of bulbs you can purchase so that the succession of bulbs produce light for at least 30 month with a probability of at least .97 ?

4. Page 83, problem 4.4

5. Page 84, problem 4.6a