1.A high volume production process produces, on average, 3.4 defective parts per hour, and the number of defects in an hour follows a Poisson distribution. Let Xi= the number of defects produced in hour i(i= 1, 2).
a) Find Pr(X1= k) for k= 0, ..., 20.
b) Assuming that X1and X2are independent, calculate Pr(X1+ X2= k) for k= 0 to 20
c) Verify numerically that Y = X1+ X2has a Poisson distribution. What are its mean and variance?
2. Consider rolling a pair of fair dice.
a) What is the probability that the number of dots on the upward side of both dice is the same (i.e., you roll (1,1) or (2,2) or (3,3) or (4,4) or (5,5) or (6,6))? [Note: we'll call this a matched roll in the remainder of this question.]
b) What is the probability that there is matched roll among the first 6 rolls?
c) Find the expectation and the standard deviation of the number of rolls you need to do until you have a matched roll (the final matched roll is included in the count)?
3. Suppose you draw a card randomly from a regular deck of cards. Let Abe the event that the card is red, and Bthe event that the card is a 9.
a) Find Pr(A), Pr(B), Pr(Aand B) and Pr(B|A).
b) Are A and B independent events? Why or why not?
Now suppose you draw a second card (without first replacing the first card in the deck). Let Cbe the event that the second card is a 9.
c) Find Pr(C), Pr(Aand C) and Pr(Band C).
d) Are Aand Cindependent events? Why or why not?
e) Are Band Cindependent events? Why of why not?
4. ABC Airlines operates 50 seat planes between Albany and Boston. For tomorrow's flight, it has accepted 62 reservations. From previous experience, it knows that each potential passenger with a reservation will show up for the flight with a probability of 0.78, independent of all other potential passengers.
a) Find the probability that exactly 50 passengers will show up.
b) Find the probability that all passengers who show up will get a seat on the flight.
c) What is the expected number of passengers who show up that will not get a seat on the flight?
d) What is the expected number of empty seats on the flight?
e) Explain how it is possible that the answers to parts c and d are both positive.
f) How many reservations can ABC accept if they want to be 98% sure that all passengers can be accommodated on the flight?
5.From New York State Department of Motor Vehicles (DMV) records, it is known that 85% of all vehicle registered in New York State have automatic transmissions, and the rest manual transmissions. It is also known that 70% of vehicles with automatic transmission are classified as passenger cars and the rest as trucks, while 45% of manual transmission cars are passenger cars and the rest trucks. Using this information, answer the following questions.
a) Find the joint distribution of vehicle type (passenger car / truck) and transmission type (automatic / manual) in New York.
b) Find the marginal distribution of vehicle type in New York.
c) Find the conditional distribution of vehicle type given that the transmission is automatic.
d) Find the conditional distribution of vehicle type given that the transmission is manual.