1. For each of the following two parts, you must justify your answer. No explanation = no credit. Partially correct explanations=partial credit.
a) True or false. For the events A and B, if (A'∩B') = Ø , then the events are exhaustive.
b) True or false. For two events A and B, if P(A)=.3, P(B)=.25, and P(AUB)=.50, then A and B are mutually exclusive.
2. A raffle is being held at a school to raise money for the library. The organizers sell 40 tickets. They will select 4 winning tickets at random for prizes.
a) How many different combinations of 4 winning tickets can there be?
b) Suppose you hold 6 tickets. What is the probability that you will win exactly 2 out of the 4 prizes.
3. Let the random variable R denote the rate of return on an asset. R has the following pdf.
|
R
|
P(R=r)
|
|
.3
|
.05
|
|
.2
|
.20
|
|
.1
|
.20
|
|
0
|
.20
|
|
-.1
|
.15
|
|
-.2
|
.10
|
|
-.3
|
.10
|
a. What is the expected value and variance of R?
b. You are considering investing $150 in this asset. After 1 year, the value will be $150(1+R). What is the expected value and standard deviation of the value of your asset in 1 year?
4. Suppose the random variable X has a mean of 16 and a variance of 144. Y is a linear transformation of the form Y=a+bX. Y has a mean of 4 and variance 36. Assuming b>0, what are the values of a and b?
5. Suppose cell phones are available in two basic types: Flashy or Dull. Furthermore, suppose cell phone owners fall into two basic categories: Hipsters or Grannies.
Let .55 be the probability of being a Hipster. Also, 40% of cell phones are both Flashy and owned by Hipsters. Finally, if a cell phone is owned by a Granny, the probability of it being Dull is .90
a) What is the probability that a cell phone is both Dull and owned by a Hipster.
b) If a phone is owned by a Hipster, what is the probability it is Flashy.
c) What is the probability that any randomly chosen phone is Dull?
6. Suppose a microchip manufacturer produces 1 defective chip for every 30 they build. They randomly select 210 chips off the assembly line for quality control testing.
a) What is the expected value and variance of the number of defective chips?
b) What is the probability that they discover exactly 9 defective chips in this test?
7. On a typical spring afternoon, customers enter a bookstore at a rate of 80 per hour.
a. What is the value of λ, defined as the average rate of customers per minute?
b. In 30 minutes, what is the a) expected value and b) standard deviation of the number of customers entering the store?
c. In a 3 minute period, what is the probability of 2 or more customers entering the store?