The U.S. Senate consists of 100 senators, with 2 from each of the 50 states. There are d

Democrats in the Senate. A committee of size c is formed, by picking a random set of

senators such that all sets of size c are equally likely.

(a) Find the expected number of Democrats on the committee.

(b) Find the expected number of states represented on the committee (by at least one

senator).

(c) Find the expected number of states such that both of the state's senators are on the

committee.

81. A certain college has g good courses and b bad courses, where g and b are positive

integers. Alice, who is hoping to find a good course, randomly shops courses one at a

time (without replacement) until she finds a good course.

(a) Find the expected number of bad courses that Alice shops before finding a good

course (as a simple expression in terms of g and b).

(b) Should the answer to (a) be less than, equal to, or greater than b/g? Explain this

using properties of the Geometric distribution.

82. The Wilcoxon rank sum test is a widely used procedure for assessing whether two groups

of observations come from the same distribution. Let group 1 consist of i.i.d. X1,...,Xm

with CDF F and group 2 consist of i.i.d. Y1,...,Yn with CDF G, with all of these r.v.s

independent. Assume that the probability of 2 of the observations being equal is 0 (this

will be true if the distributions are continuous).

After the m + n observations are obtained, they are listed in increasing order, and each

is assigned a rank between 1 and m + n: the smallest has rank 1, the second smallest

has rank 2, etc. Let Rj be the rank of Xj among all the observations for 1 ? j ? m,

and let R = Pm

j=1 Rj be the sum of the ranks for group 1.

Intuitively, the Wilcoxon rank sum test is based on the idea that a very large value of

R is evidence that observations from group 1 are usually larger than observations from

group 2 (and vice versa if R is very small). But how large is "very large" and how small

is "very small"? Answering this precisely requires studying the distribution of the test

statistic R.

(a) The null hypothesis in this setting is that F = G. Show that if the null hypothesis

is true, then E(R) = m(m + n + 1)/2.

(b) The power of a test is an important measure of how good the test is about saying

to reject the null hypothesis if the null hypothesis is false. To study the power of the

Wilcoxon rank sum stest, we need to study the distribution of R in general. So for this

part, we do not assume F = G. Let p = P(X1 > Y1). Find E(R) in terms of m, n, p.

Hint: Write Rj in terms of indicator r.v.s for Xj being greater than various other.

Let X1, X2, ..., X70 be an iid random sample of size n = 70 from a distribution with p.d.f. f(z) = =(6 -x)2, 070 a. Find P(X 0 the function 0 if 1: 12). ((1) Suppose X is a random variable with probability density function as in part (a). If Pr(10 g X g 20) = 828:1,what is the mean ofX