Name:

Statistics 510

Fall 2005-Final

Instructions

ü  This exam is closed book.

ü  You may use four pages of notes (double sided). You may use a calculator.

ü  A normal table is attached to the front of the exam.

ü  Two sheets of paper are attached to the back of the exam for overflow. Scratch paper is available at the front of the room.

ü  The points for each question are marked on the exam. The parts of each question will be weighted equally.

ü  Time = 120 minutes.

ü  Good luck!

Question / Max. Score / Score Received
1 / 10
2 / 15
3 / 10
4 / 10
5 / 25
6 / 10
7 / 15
8 / 15
Total Score / 110


1.(10) Two litters of a particular rodent species have been born, one with two-brown haired and one gray-haired rodent (litter 1), and the other with three brown-haired and two gray-haired rodents (litter 2). We select a litter at random and then select a rodent at random from the selected litter.

(a) What is the probability that the rodent chosen is brown-haired?

(b) Given that a brown-haired rodent was selected, what is the probability that the sampling was from litter 1?

2.(15) A certain river floods every year. Suppose that the low-water mark is set at 1 and the high-water mark Y has cumulative distribution function

(a) Find the probability distribution function of Y.

(b) Find the expected value of Y.

(c) If the low-water mark is reset at 0 and we use a unit of measure that is of that given previously, the high-water mark becomes . Find the cumulative distribution function .

3.(10) An ambulance travels back and forth, at a constant speed along a road of length 1 mile. At a certain moment of time an accident occurs at a point on the road. The probability density function of the location of the accident is

Assuming that the ambulance’s location at the moment of the accident is uniformly distributed on [0,1] and that the ambulance’s location is independent of the accident’s location, compute the probability density function for the ambulance’s distance from the accident.

4.(10) A number is chosen from the uniform distribution on the interval (0,3). A second number is chosen independently from the uniform distribution on the interval (0,4). What is the eightieth percentile of the sum of the two numbers? [By definition, the 80th percentile, , is that number for which ].

5.(25) Let cost to make a movie (in 100 million dollar units)

the revenue from a movie (in 100 million dollar units)

Assume that

(a) Find the marginal probability density function of X, .

(b) Find the moment generating function of and use the moment generating function to find .

(c) If a movie cost 100 million dollars to make , what is the expected value for the revenue.

(d) The profit for the movie is . Find E(profit).

(e) What is the probability that the profit is greater than zero?

6.(10) Coupons in cereal boxes are numbered 1 to 5, and a set of one of each is required for a prize. Each time you obtain a cereal box, the coupon in it is equally likely to be any of the 5 numbers. Let B be the number of cereal boxes you need to collect to obtain a complete set of coupons with each of the 5 numbers.

(a) Find the probability that .

(b) Find .

7.(15) Consider two random variables and . Suppose the distribution of Y, conditional on , is normal with mean x and variance and that the marginal distribution of X is uniform on the interval (0,1).

(a) Find .

(b) Find

(c) Find .

8.(15) An Airline “overbooks” a flight because it expects that there will be no-shows. Assume that

(i) There are 200 seats available on the flight.

(ii) Seats are occupied only by individuals who made reservations (no standbys).

(iii) The probability that a person who made a reservation shows up for the flight is 0.95.

(iv) Reservations show up for the flight independently of each other.

(a) If the airline accepts 220 reservations, write an expression for the exact probability that the plane will be full (i.e., at least 200 reservations show up). You do not need to evaluate this expression.

(b) Use the central limit theorem to approximate the probability in (a).

(c) Suppose the airline wants to choose a number n of reservations so that the probability that at least 200 of the n reservations show up is 0.75. Find the (approximate) minimum value of n.