Download Probability (Math 336)

Probability (Math 336) - Exam #2
Fall 2000 - Hartlaub
You may use the properties and results we have proven in class and in homework exercises
without proof, unless the problem specifically asks for a proof of a particular result. Good Luck
and enjoy your break!
Part 1 - Your Choice: Solve either problem 1 or problem 2, but not both.
1.
(10)
(5)
(10)
2.
(10)
(5)
(10)
Let Sn = X1 + X2 + ... + Xn with each independent Xi following a Poisson(i) distribution.
a. Find the probability generating function for Sn.
b. Use the probability generating function in part (a) to find the mean of Sn.
c. Use the probability generating function in part (a) to find the variance of Sn.
Using a standard deck of cards, the system of bridge bidding assigns points as follows:
X(ace) = 4, X(king) = 3, X(queen) = 2, X(jack) = 1, and X() = 0 for any other card .
a. Find the probability generating function for X.
b. Use the probability generating function in part (a) to find the mean of X.
c. Use the probability generating function in part (a) to find the variance of X.
Part 2 - Solve Problem 3
3.
(10)
(10)
(10)
A particle counter records two types of particles, Types 1 and 2. Type 1 particles arrive at
an average rate of 1 per minute, Type 2 particles arrive at an average rate of 2 per minute.
Assume these are two independent Poisson processes. Give numerical expressions for the
following probabilities:
a. Three Type 1 particles and four Type 2 particles arrive in a two-minute period;
b. The total number of particles of either type in a two-minute period is 5;
c. The fourth particle arrives in the first five minutes.
Part 3 - Pick 5: Solve 5 of the remaining 6 problems. Clearly identify the five problems you
have chosen. Each problem is worth 20 points.
4.
A random variable has a probability density function of the form
cx 2 , 0  x  1
f ( x)  
0, otherwise.
(10)
(10)
a. Find the constant c.
b. Find P( X  a ) for 0  a  1.
5.
The lifetime (in hours) Y of an electronic component is a random variable with density
function given by
y
 1 100
e , y0

f ( y )  100
0, otherwise.

Three of these components operate independently in a piece of equipment. The equipment
fails if at least two of the components fail. Find the probability that the equipment will
operate for at least 200 hours without failure. (20 points)
6.
(10)
(10)
7.
If a parachutist lands at a random point on a line between markers A and B, find the
probability that
a. she is closer to A than to B; (Identify your probability model and justify your answer.)
b. her distance to A is more than three times her distance to B. (Justify your answer.)
A gas station operates two pumps, each of which can pump up to 10,000 gallons of gas in a
month. The total amount of gas pumped at the station in a month is a random variable Y
(measured in 10,000 gallons) with a probability density function given by
 y, 0  y  1

f ( y )  2  y , 1  y  2
0, elsewhere.

(10)
(10)
8.
a. Find the probability that the station will pump between 8,000 and 12,000 gallons in a
particular month.
b. Given that the station pumped more than 10,000 gallons in a particular month, find the
probability that the station pumped more than 15,000 gallons during this month.
For certain ore samples the proportion Y of impurities per sample is a random variable with
density function given by
3 2
 y  y, 0  y  1
f ( y)   2

0, elsewhere.
The dollar value of each sample is W  5  0.5Y . Find the mean and variance of W. (20)
9.
As you know, the memoryless property holds for one discrete distribution and one
continuous distribution. State and prove the memoryless property for one of the two
distributions. (20)