Math 308 Test 3 Carter Name_________________________
Show all work in order to receive credit. 11/9/2001
1. Given f(y), find the value of k which makes f(y) a valid density function. (8)
.,0
10),1(
)(
2
elsewhere
yyky
yf
2. Let
.,0
11),2(
4
1
)(
elsewhere
xx
xf
.
(20)
a) Find E[X].
b) Find the distribution function of X.
c) Find P(0 < X < 0.5)
d) Find P( X > 0.5 )
3. A morning news program that airs from 7:00 a.m. until 9:00 a.m. weekdays has 5-minute local segments at 7:25, 7:55,
8:25, and 8:55. If you randomly tuned in to the program, what is the probability that the local segment is airing? (8)
4. Let X denote a normal random variable with a mean of 100 and a standard deviation of 40. What is the probability that X
is between 144 and 170? (8)
5. The time intervals between dial-up connections to a computer center from remote terminals are exponentially distributed,
with mean of 15 seconds.
a) Find the probability distribution of the waiting time from the opening of the computer center until the fifth dial-up
connection from a remote terminal.
b) What is the probability that that at least 1.5 minutes elapse before the fifth dial-up connection? (Setting up the
appropriate integral is sufficient.) (12)
6. A manufacturer of commercial chemicals sells a particular product in steel drums. The net weights of the product in these
drums have a normal distribution with mean 310 pounds and standard deviation of 4 pounds. The manufacturer guarantees
that each drum will contain at least 300 pounds of this product. (16)
a) What percentage of the drums will satisfy this guarantee?
b) If three drums are randomly selected, what is the probability that exactly two satisfy the guarantee?
7. Define m(t), the moment-generating function of a discrete random variable, X. What is a use of a moment generating
function? (8)
8. Suppose Y has a Beta distribution with = 5 and If Q = 16Y+5, find the mean and variance of Q. (8)
9. Select one of the following to prove: (10)
a) If X has an exponential distribution with mean theta, then P(X > a+b | X > a) = P ( X > b).
b) If
0
x1α
dxexΓ(α)
, then (a+1) = a(a).
c) Derive the mean of the Beta distribution
