# Probability

Find the probability for the various situations

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 o on the same floor? [2]

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

(c) What is the probability that all 5 individuals will get o on 3 diferent 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��1p: x = 1; 2;

(a) Find MX(t), the moment-generating function of X. [4]

(b) Use your result in (a) to nd 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 prot, 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 prot. [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, Je, Mike, and Wai play a game where this unbalanced dice is tossed until the rst 6

appears. The guy who tosses the rst 6 wins, the prize being that the losers will pay the winner's portion

of July's rent. Je, being the oldest, tossess rst, followed by Wai and then Mike. This sequence continues

until the rst 6" appears.

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

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

1For example, a 6" is six times as likely as a 1", a 5" is ve 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 rst guest will arrive within the rst 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 nite mean and a nite variance 2,

with a moment generating function

MXi (t) = e1:5t+3t2

(a) What is the distribution of each Xi [2]

(b) Compute the moment generating function of X =

P10

i=1

Xi

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]

#### Solution Preview

Please see the attachment.

1. (a) What is the probability that all 5 individuals will get off on the same floor?

Solution. Since if 5 individuals will get off on the same floor, there are 9 possible cases. The total possible case is since each person has 9 possible choices to get off. So, the probability is

(b) What is the probability that all 5 individuals will get off on the two different floors?

Since if 5 individuals will get off on the two different floors, there are possible cases. The total possible case is since each person has 9 possible choices to get off . So, the probability is

(c) What is the probability that all 5 individuals will get off on the 3 different floors?

Since if 5 individuals will get off on the 3 different floors, there are possible cases. The total possible case is since each person has 9 possible choices to get off. So, the probability is

2. (a) exceed 9.85%

Solution. Denote the prime interest rate by X, then X follows normal distribution . ...

#### Solution Summary

Probability in various situations are found. The moment of generating a function is computed.