Math 309Test 2CarterName______
Show all work in order to receive credit.10/8/01
distributionbinomialgeometricnegative binomialPoissonhypergeometric
meannp1/pr/pn*(k/N)
variancenpqq/p2rq/p2
1. The number of cars abandoned on a certain highway averages 2.2 per week.
a)What is the probability that no car is abandoned on the highway next week?
b)What is the probability that at least 3 cars are abandoned in the next two weeks?
2. Y is a random variable with mean 500 and variance 100. Find a, b, c, d such that:
a)P ( a < Y < b ) ¾
b)P ( c < Y < d ) 8/9
- Daily sales records for a computer manufacturing firm show that it will sell zero, one, or two mainframe computer systems with the following probabilities:
Number of sales 0 1 2
Probability0.70.20.1
a)Find the expected value, the variance, and standard deviation for daily sales.
b)Find the cumulative distribution function (cdf) for the number of sales. Sketch its graph.
4. A multiple-choice quiz has 5 questions each with 3 possible answers. There is one correct answer for each question. A student who hasn’t studied randomly guesses on each question.
a)Find the probability that he gets exactly 4 of the questions correct.
b)Find the probability that he gets at least 4 of the questions correct.
5. A jar holds 7 chocolate chip cookies, 5 peanut butter cookies, and 3 oatmeal cookies. Janet grabs 4 cookies. Let X denote the number of chocolate cookies selected.
a)P( X 1)
b)P( X = 3 )
c)P( X = 3 | X 1)
6. A recruiting firm finds that 30% of a large pool of applicants have the required training for a certain position. The firm conducts interviews sequentially and randomly.
a)Find the probability that it takes exactly ten interviews to find four qualified applicants.
b)Let Y denote the number of interviews required to find 4 qualified applicants, find the mean and variance of Y.
c)Would you be surprised if it took 18 interviews to find four qualified applicants? Why or why not?
7. Let X denote a binomial random variable with n = 10, p = 0.4. Find the mean and variance of C if C = 3x + 5.
8. Suppose that 8% of the engines manufactured on a certain assembly line are defective. Engines are randomly selected one at a time and tested.
a)Find the probability that the first defective is found on the 6th test.
b)Find the probability that more than ten engines are tested before finding a defective.
c)If you have tested three engines and have found no defective, what is the probability that you will test more than 13 before finding a defective?
9. Prove two of the following:
a)Prove that if Y is a binomial random variable, E[Y] = np; or if Y is a Poisson random variable, E[Y] = .
b)Prove that if Y = aX + b, then V(Y) = a2V(X) for constants a and b.
c)Prove that if X is a geometric random variable and c > b, then P(X > c | X > b) = P(X > c-b).
[You may use P( X > j) = qj, j = 0, 1, 2, 3, . . .]
Answers:1a. e^(-2.2), b. 1-e-4.4(1+4.4+(4.4)2/2);
2. a=480, b=520, c=470, d=530;
3. 0.4, .6-.42, (.44).5; 0 for x < 0, .7 for 0<= x < 1, 0.9 for 1 <= x < 2, 1 for x >= 2;
4. 10/243, 11/243;
5. 259/273, 280/1365, 280/1295;
6. .0800; ; not surprising, 18 is within 1 standard deviation of the mean;
7. E[C] = 17,
8. (.92)5(.08), (.92)10, (.92)10