**Exercise 2.5**: If is a random variable with , show that the probability that is even is .

.

Since only depends on , all we need to prove is : .

Indeed, we have .

**Exercise 2.6**: Suppose that we independently roll two standard six-sided dice. Let be the number that shows on the first dice, the number on the second dice, and X the sum of the numbers of two dice.

**(a) **

**(b) **

.

**(c) **

.

**(d) **.

**Exercise 2.7: **Let and be independent geometric random variables, where has parameter and has parameter .

**(a) **

**(b) **

First we will calculate and .

.

.

Because is a discrete random variable takes on only nonnegative integer value, we have:

.

**(c) **

.

**(d) **

**Exercise 2.13**:

**(a) **Consider the following variation of the coupon collector’ s problem. Each box of cereal contains one of different coupons. The coupons are organized into pairs, so that coupons 1 and 2 are a pair, coupons 3 and 4 are a pair, and so on. Once you obtained one coupon from every pair, you can obtain a prize. Assuming that the coupon in each box is chosen independently and uniformly at random from the possibilites, what is the expected number of boxes you must buy before you can claim the prize?

**(b) **Generalize the result of the problem in part (a) for the case where there are different coupons, organized into disjoint sets of coupons, so that you need one coupon from every set.

**(a) **Let be the total number of boxes we must buy before we have at least one coupon from every pair, and the number of boxes we buy until we get a coupon from th pair when we already had coupons from exactly pair.

Certainly we have .

The probability to get a coupon from a new pair when already had coupons from pairs is

Since each is a geometric random variable with parameter , their expected values are: .

Thus .

Use the same technique described in the text, we can prove .

**(b) **Things are essentially the same when we have coupons in different sets. The expected number of boxes we must buy before having at least one coupon from every set is .

Posted by nitesh on April 7, 2012 at 11:53 am

If you have solution for 2.10 please let me know