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Math 309
Test 2
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distribution
mean
variance
binomial
np
npq
Carter
geometric
1/p
q/p2
Name__________________________________
10/8/01
negative binomial
r/p
rq/p2
Poisson


hypergeometric
n*(k/N)
k  N  n 
 k 
n 1  

N  N  1 
 N 
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
3.
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
Probability
0.7
0.2
0.1
a)
b)
Find the expected value, the variance, and standard deviation for daily sales.
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 6 th 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;
4(.7)
4
 13.3,  2  2  31. 1 ; not surprising, 18 is within 1 standard deviation of the mean;
.3
.3
2
7. E[C] = 17,  C  21.6
6. .0800;

8. (.92)5(.08), (.92)10, (.92)10