MATH 411, PRACTICE TEST 2
Q1. A random variable X has density

 x , 1 ≤ x ≤ 3,
fX (x) = 4
0, otherwise.
Find the distribution function of the random variable
1
Y =√ .
X
Q2. A town has two newspapers, the Post and the Chronicle. Every year
30% of Post readers switch to the Chronicle and 20% of Chronicle readers
switch to the Post. If the current vector is (.5, .5), what will it be in 2 years
time? Also find the stationary distribution for this Markov process.
Q3. X and Y have joint density function

2x + 2y, 0 ≤ y ≤ x ≤ 1,
fX,Y (x, y) =
0,
otherwise.
Find the marginal density of X.
Q4. X has uniform density on [0, 1]. Find the density function for Y = eX
and find the expected value of Y .
Q5. Let X and Y have joint density

2y, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1,
fX,Y (x, y) =
0, otherwise.
Let Z = X + Y . For z in the interval [0, 1], find P (Z ≤ z).
Q6. Xavier and Yolanda plan to meet for lunch between noon and 1 p.m.
They arrive independently with uniform distribution on [0, 1]. Yolanda will
wait 30 min. for Xavier, but Xavier will only wait 15 min. for Yolanda.
What is the probability that they meet? (write the conditions as inequalities
and draw a picture)
Q7. Al and Betty are each working independently on a project starting at
the same time. The time to completion for Al is uniformly distributed on
[0, 2] hours and for Betty is uniformly distributed on [0, 1] hours. They will
go home together when both have finished. Let T be the time to when they
have both finished. Find the density function for T and find the expected
value of T .
Q8. Every Saturday Jim and Mary either play tennis, golf, or go jogging.
They never do the same activity two consecutive weeks. After a tennis week,
or a golf week they switch to one of the other activities with equal probability.
After a jogging week they switch to tennis 70% of the time. In the long run,
what percentage of their Saturdays do they devote to golf?
Q9. An urn contains two red balls labeled 1, 2, and three blue balls labeled
1,2,3. A ball is picked (but not replaced) and X is the number on it. A second
ball is then picked and Y is the number on it. Find the joint distribution for
X and Y .
Q10. Fill in the rest of the joint distribution so that X and Y are independent. There is more than one possible answer so find them all.
Y
X=0
1
0
?
3
16
1
3
16
?