Probability

Sample Final Exam - From Spring 2001

1. 5 individuals enter the ground floor of an elevator of a building which has 10 floors(9 floors above the
ground floor). Assuming that each of the 5 individuals is going to depart the elevator on one of the 9 floors
above the ground floor,

(a) what is the probability that all 5 individuals will get off on the same floor? [2]




(b) what is the probability that all 5 individuals will get off on 2 different floors? [3]




(c) What is the probability that all 5 individuals will get off on 3 different floors? [3]
2. One method of arriving at economic forecasts is to use a consensus approach. A forecast is obtained
from each of a large number of analysts; the average of these individual forecasts is the consensus forecast.
Suppose that the individual 1998 January prime-interest-rate forecasts of all economic analysts are
approximately normally distributed with a mean of 7.1% and a standard deviation of 2.65%. If a single
analyst is randomly selected from among this group, what is the probability that the analyst's forecast of
the prime interest rate will

(a) exceed 9.85% [4]




(b) 10 analysts from this group are randomly selected. What is the probability that at most 8 of these
analysts forecasted the prime interest rate to be less than 9%. [4]
3. A random variable X having a geometic distribution has the following probability function

P (X = x) = (1 - p)x-1 p. x = 1, 2, · · ·



(a) Find MX (t), the moment-generating function of X. [4]
(b) Use your result in (a) to find the mean of X and the variance of X. [4]
4. A tool and die company makes casting for steel stress-monitoring gauges. Their annual profit, Q (in
$100,000's), can be expressed as a function of demand (X):

Q(x) = 3(1 - e-3x ).

Suppose the total demand (in 1000's) for their castings follows the probability distribution with density:

f (x) = 5e-5x for x > 0.

(a) What is the probability that the total demand for castings is between 2000 and 5000? [3]




(b) Find the company's expected profit. [5]
5. A dice is unbalanced in such a way that the probability of the dice showing "i" dots is proportional to
how many dots on the dice1 .

Three roommates, Jeff, Mike, and Wai play a game where this unbalanced dice is tossed until the first 6
appears. The guy who tosses the first 6 wins, the prize being that the losers will pay the winner's portion
of July's rent. Jeff, being the oldest, tossess first, followed by Wai and then Mike. This sequence continues
until the first "6" appears.

(a) How many tosses can be expected to occur until the first "6" appears? [2]




(b) What is the probability that Jeff will not have to pay next month's rent? [6]




1
For example, a "6" is six times as likely as a "1", a "5" is five times as likely as a "1", etc.
6. Guests arriving at a hotel in accordance with a Poisson process, at a rate of 5 per hour.

(a) What is the probability that no one will arrive in the next hour? [2]




(b) What is the probability that the first guest will arrive within the first 10 minutes? [3]
(c) How much time (in minutes) should the hotel expect to pass until the fourth guest arrives from the top
of the hour? Provide a measure of dispersion as well. [3]
7. Let X1 , X2 , · · · , Xn be a random sample from a population with a finite mean µ and a finite variance 2 ,
with a moment generating function
2
MXi (t) = e1.5t+3t
(a) What is the distribution of each Xi [2]




10
Xi
(b) Compute the moment generating function of X = i=1
10 and identify the distribution of X. [3]
8. A random sample of size 2 is available from a Poisson distribution with parameter . Let X denote the
sample mean and let Y = (6X1 - 3X3 )/10.

Are Y and X unbiased estimators for ? Is so, which of the two estimators would you recommend. Explain. [8]