In Fall 2011, We Covered #1-5 on Test 1, but You Should Be Able to Do All of These Problems

In Fall 2011, we covered #1-5 on Test 1, but you should be able to do all of these problems.

Math 309 Test 2 Carter Name______

Show all work in order to receive credit.

1.  A Weather Service predicts that for tomorrow there is a 50% chance of tornadoes, a 20% chance of a monsoon, and a 10% chance of both. If the predictions are accurate, what is the probability that we encounter neither tornadoes nor a monsoon tomorrow? (6)

2.  A manufacturer of lie detectors determines that its newest design has a probability of 0.1 of saying that someone has lied if he/she really told the truth, and a probability of 0.2 of saying that someone has told the truth if he/she really lied. On a given question, 40% of the population lies.

a)  What is the probability that the machine says that a person lied if he/she did lie?

b) What is the probability that the machine says that the person lied? (12)

3.  An electric circuit has four relays (see diagram). Current will flow from point A to point B if there is at least one closed path when the relays are activated. Suppose that the relays act independently of one another and close properly when activated with a probability of 0.8. Find the probability that the current flows when the relays are activated. (8)

A B

4. A jug contains 10 black marbles and 6 white marbles. You randomly select 6 without replacement.

a) What is the probability that you have 3 marbles of each color?

b) What is the probability that you have at least 1 black marble? (12)

5. The table gives the probability distribution of the winnings when you play a game: x $5 $2 $1 $0

p(x) .3 .2 ? .1

a) Find P(X=1) = ______

b) Find the mean and variance of X. (10)

6.  Roll a die until you get a three 5’s. Let Y be the number of rolls. Find P(Y = 7). (6)

7. Y defines a binomial random variable where n = 40 and p = 0.3. Is it likely that y = 21? Justify your response. (8)

8. Suppose that Y is a random variable with mean 5 and variance 4. Let X = 3Y + 10. Find the mean and variance of X. (8)

9.  Two construction contracts are to be randomly assigned to one or more of three firms: I, II, and III. Any firm can receive both contracts. Each contract will yield a profit of $90,000 for the firm. Let Y represent the number of contracts firm I receives. Find the probability distribution for Y. (10)

10. The probability that a certain student pilot passes the written test for his private pilot's license is 0.3. Let Y represent the number of times that a student takes the test before passing.

a) Find the probability that he passes on his 4th try.

b) What is the probability that it takes him more than 4 tries?

c) He has already taken the test 4 times, what is the probability that he takes more than 6 tries to pass the test? (12)

11. Choose one of the following to prove.

a) If Y has a binomial distribution, m= np

b) If Y has a geometric distribution, .

c) If Y is geometric, then for a positive integer a, P(Y > a) = qa. (8)
______Answers: 1) 0.4 2) 0.8, 0.38 3) 0.8704 4) 0.2997, 0.99988 5) 0.4, 2.3, 3.41 6) 0.0335 7) no, 21 is more than 3 s.d. from the mean 8) 25, 36 9) p(0)=4/9 p(1)=4/9, p(2)=1/9

10) (.7)3*(.3), (.7)4, (.7)2 11) see class notes

Math 309 Test 3 Carter Name______

Show all work in order to receive credit.

1. Service calls come to a maintenance center according to a Poisson process and on the average 2.7calls come per minute.

a) Find the probability that no more than 4 calls come in any one minute.

b) Find the probability that more than 10 calls come in a 5-minute period. (10)

© is an exponential r.v. – we didn’t derive this in2011, I’ll do it after the test.

c) Suppose that you know that one call came in a one-minute period. What is the probability that it came in the first 15 seconds? (5)

2. Find the value of the constant, c, so that f(y) is a density function. (10)

3a) Find F(x), given f(x). (10)

b)  Find P( 1 < X < 2 ).

Problems 4- 6 concern material that is not covered for the test 2 during Fall 2011.

4. Given m(t) = 0.3et + 0.2e2t + 0.35e4t + 0.15e7t (10)

a)  Find probability distribution, p(Y).

b)  Find P(Y ³ 2 ).

5. Use the definition to derive the moment generating function for either a Poisson or a Geometric random variable. (10)

6. Use the moment generating function to show that the mean of a binomial random variable is np. (10)

7. A manufacturer has contracted to supply ball bearings. Product analysis reveals that the diameters are normally distributed with a mean of 25.1 mm and a standard deviation of 0.2 mm. (10)

a)  What is the probability that a diameter exceeds 25.4 mm?

b)  What is the minimum acceptable diameter if the smallest 13% of the diameters are unacceptable?

8.  The proportion of time that an industrial robot is in operation during a 40-hour week is a random variable with probability density function .

a)  Find the expected value and variance of Y. (15)

b)  If the profit for a week is given by X = 200Y – 60, find E(X) and V(X).

9a) Is the distribution function at the right for a discrete or

continuous random variable?

b) Find the density function or probability distribution of X.

(10)

Answers:

1) 0.8629, 0.7888 2) 3/16

3) 0 if X < -2; x3/16 + ½ if –2 £ X £ 2; 1 if X > 2 ; 7/16

7) 0.0668, 24.874 8) 2/3, 1/18; 73.333, 2222.222

9) discrete; p(1)=0.2, p(2)=0.5, p(4)=0.3