The distributions of the grades of the students of

Show that if X is a normal random variable with parameters (, 2), then for R, we have that MX(t) = exp 4 t + (1/2)22t 2 5 .

Let X1, X2, . . . , Xn be independent geometric random variables each with parameter p. Using moment-generating functions, prove that X1 + X2 + + Xn is negative binomial with parameters (n, p).

Let X1, X2, . . . , Xn be n independent exponential random variables with the identical mean 1/. Use moment-generating functions to find the probability distribution function of X1 + X2 + + Xn.

Using moment-generating functions, show that the sum of n independent negative binomial random variables with parameters(r1, p),(r2, p), . . . , (rn, p)is negative binomial with parameters (r, p), r = r1 + r2 + + rn.

Let X1, X2, . . . , Xn be n independent gamma random variables with parameters (r1, ), (r2, ), ... , (rn, ), respectively. Use moment-generating functions to find the probability distribution function of X1 + X2 + + Xn.

The probability is 0.15 that a bottle of a certain soda is underfilled, independent of the amount of soda in other bottles. If machine one fills 100 bottles and machine two fills 80 bottles of this soda per hour, what is the probability that tomorrow, between 10:00 A.M. and 11:00 A.M., both of these machines will underfill exactly 27 bottles altogether?

Let X and Y be independent binomial random variables with parameters (n, p) and (m, p), respectively. Calculate P (X = i | X + Y = j ) and interpret the result.

LetX, Y , andZ be three independent Poisson random variables with parameters 1, 2, and 3, respectively. For y = 0, 1, 2, . . . , t, calculate P (Y = y | X+Y +Z = t).

Mr. Watkins is at a train station, waiting to make a phone call. There is only one public telephone booth, and it is being used by someone. Another person ahead of Mr. Watkins is also waiting to call. If the duration of each telephone call is an exponential random variable with = 1/8, find the probability that Mr. Watkins should wait at least 12 minutes before being able to call.

Let X N (1, 2) and Y N (4, 7) be independent random variables. Find the probability of the following events: (a) X + Y > 0, (b) X Y < 2, (c) 3X + 4Y > 20.

The distribution of the IQ of a randomly selected student from a certain college is N (110, 16). What is the probability that the average of the IQs of 10 randomly selected students from this college is at least 112?

Vicki owns two department stores. Delinquent charge accounts at store 1 show a normal distribution, with mean $90 and standard deviation $30, whereas at store 2 they show a normal distribution with mean $100 and standard deviation $50. If 10 delinquent accounts are selected randomly at store 1 and 15 at store 2, what is the probability that the average of the accounts selected at store 1 exceeds the average of those selected at store 2?

Let the joint probability density function of X and Y be bivariate normal. Prove that any linear combination of X and Y , X + Y , is a normal random variable. Hint: Use Theorem 11.7 and the result of Exercise 6, Section 10.5.

Let X be the height of a man, and let Y be the height of his daughter (both in inches). Suppose that the joint probability density function of X and Y is bivariate normal with the following parameters: X = 71, Y = 60, X = 3, Y = 2.7, and = 0.45. Find the probability that the man is at least 8 inches taller than his daughter. Hint: Use the result of Exercise 13.

The capacity of an elevator is 2700 pounds. If the weight of a random athlete is normal with mean 225 pounds and standard deviation 25, what is the probability that the elevator can safely carry 12 random athletes?

The distributions of the grades of the students of probability and calculus at a certain university are N (65, 418) and N (72, 448), respectively. Dr. Olwell teaches a calculus section with 28 and a probability section with 22 students. What is the probability that the difference between the averages of the final grades of these two classes is at least 2?

Suppose that car mufflers last random times that are normally distributed with mean 3 years and standard deviation 1 year. If a certain family buys two new cars at the same time, what is the probability that (a) they should change the muffler of one car at least 11 2 years before the muffler of the other car; (b) one car does not need a new muffler for a period during which the other car needs two new mufflers?

An elevator can carry up to 3500 pounds. The manufacturer has included a safety margin of 500 pounds and lists the capacity as 3000 pounds. The buildings management seeks to avoid accidents by limiting the number of passengers on the elevator. If the weight of the passengers using the elevator is N (155, 625), what is the maximum number of passengers who can use the elevator if the odds against exceeding the rated capacity (3000 pounds) are to be greater than 10,000 to 3?

Let the joint probability mass function of X1, X2, . . . , Xr be multinomial, that is, p(x1, x2, . . . , xr) = n! x1! x2! xr! px1 1 px2 2 pxr r , where x1 + x2 + + xr = n, and p1 + p2 + + pr = 1. Show that for k < r, X1 + X2 + + Xk has a binomial distribution.

Kim is at a train station, waiting to make a phone call. Two public telephone booths, next to each other, are occupied by two callers, and 11 persons are waiting in a single line ahead of Kim to call. If the duration of each telephone call is an exponential random variable with = 1/3, what are the distribution and the expectation of the time that Kim must wait until being able to call?