A hacker is trying to break into a password-protected website by randomly trying to

guess the password. Let m be the number of possible passwords.

(a) Suppose for this part that the hacker makes random guesses (with equal probability),

with replacement. Find the average number of guesses it will take until the hacker guesses

the correct password (including the successful guess).

(b) Now suppose that the hacker guesses randomly, without replacement. Find the average number of guesses it will take until the hacker guesses the correct password (including

the successful guess).

Hint: Use symmetry to find the PMF of the number of guesses.

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.

A small company in Montreal manufactures sheet metals. Their daily production is 100 square meters. The number of defective spots detected in a run of 100 square meters of sheet metal (i.e. the daily production of the company) is a Poisson random variable with parameter 1 =3. A young engineer who is recently employed in the company develops a new modified manufacturing process which reduces the number of defective spots detected in 100 square meters of sheet metal to a Poisson random variable with parameter 2 =2 with a 75% probability. The company tries the young engineer's modified process in one day's production (i.e. 100 square meters of sheet metal) and observes no defective spots for that day. What is the probability that the young engineer's modified process beneficial to the company? (Show your solution, show your work.)20 6 Question 15 Not yet answered Marked out of 1.00 Flag question In the simple circular-flow diagram, markets consist of: Select one. a. the market for goods and services and the market for factors of production b. the market for goods and services and the financial market c. the market for goods and services Od. the market for the factors of production6. (22pts)The mean hourly pay rate for financial managers in the East North Central region is $32.62, and the standard deviation is $2.32 (Bureau of Labor Statistics, September 2005). Assume that pay rates are normally distributed. For a randomly selected financial manager a. What is the probability the manager earned less than $27 per hour? b. What is the probability a financial manager earns more than $ 35 per hour? c. What is the probability a financial manager earns between $27 and $35 per hour