Statistics 2015 Exercise class Week 5
Problem 1:
Out of 9 executives in a certain business rm, four are married, three have never married
and two are divorced. Three of the executives are selected at random for promotion. Let
Y1 denote the number of selected married executives, Y2 the number of selected never
married executives.
i) Find the joint distribution of Y1 and Y2 .
ii) Find the marginal distribution of Y1 .
Problem 2:
Let the distribution of Y1 and Y2 be given by the following table
Y2 \Y1
0
1
2
0
1
2
1
9
2
9
1
9
2
9
2
9
1
9
0
0
0
i) Find the marginal distributions of Y1 and Y2 .
ii) Is Y1 independent of Y2 ?
Problem 3:
If Y1 is the total time between a customer's arrival in the store and leaving the service
window and if Y2 is the time spent in a line before reaching the window, the joint density
of these variables is
(
f (y1 , y2 ) =
e−y1
0
0 ≤ y2 ≤ y1 < ∞
.
elsewhere
Are Y1 and Y2 independent?
Problem 4:
The following function is proposed as a cumulative distribution function for the bivariate
random variable (X, Y )
(
(1 + e−(x/10+y/20) − e−x/10 − e−y/20 ) x > 0, y > 0
FX,Y (x, y) =
0
else.
i) Verify that the function has the appropriate properties to serve as a cumulative
distribution function.
ii) Find the marginal density and cdf of Y .
iii) What is the probability that X ≤ 10 and Y ≤ 20.
iv) Are X and Y independent random variables?
Problem 5:
The waiting time Y until delivery of a new component for an industrial operation is
uniformly distributed over the time interval from 1 to 5 days. The cost of this delay is
given by U = 2Y 2 + 3. Find the density function of U .
Problem 6:
A physical tness test was given to a large number of college freshmen. In part of the
test, each student was asked to run as far as he or she could in 10 minutes. The distance
each student ran in miles was recorded and can be considered a random variable, say X .
The data showed that the random variable X followed the log-normal distribution with
µy = 0.35 and σy = 0.5, where Y = ln X . A student is considered physically t if he
or she is able to run 1.5 miles in the time allowed. What percentage of college freshmen
would be considered physically t if we consider just this part of the test?