PSTAT160A - Introduction to Stochastic Processes Brunick
PSTAT160A - Midterm - Solutions
Please write your name and section time on your solutions and then solve the following
problems. Please start each new problem (but not subproblem) on a new sheet of paper. To
receive full credit, you must label the relevant events and random variables and justify all the
steps in your calculation. Useful justifications include “by independence,” “by the law of total
probability,” “by the tower law of conditional expectation,” etc. Err on the side of providing too
many details.
1. (20 pts) Suppose that we place two white balls in an empty urn. We then place a number of black
balls in the urn where the number black balls is determined by the roll on a fair die. After this
procedure, there are between three and eight balls total in the urn. Finally, we select a ball from
the urn. Assume that all balls are equally likely to be selected in this last step. Compute the
probability that we select a white ball conditional on the fact that the roll of the die is even. Please
fully simplify your answer.
Solution: Let Wdenote the event that the ball selected is white, and let Xdenote the number
rolled on the die. The problem tells us that P(W|X=i) = 2/(2 + i), so
P(W|Xis even) = P(Wand Xis even)
P(Xis even)
=P3
i=1 P(W|X= 2i)P(X= 2i)
1/2(by law of total probability)
=2/4·1/6+2/6·1/6+2/8·1/6
1/2= 13/36.
2. (20 pts) Let Ube uniformly distributed on (0,1) and let Xbe a geometric random variables with
parameter p, so P(X=i) = pqi−1for i∈Nwhere q= 1 −p. Now assume that Uand Xare
independent and compute P(U≤1/X!), where as usual, the “!” denotes the factorial. Please
simplify your answer. It may help to recall that ex=Pn∈N0xn/n!.
Solution: We have
P(U≤1/X!) = X
i∈N
P(U≤1/i!|X=i)P(X=i) (by law of total probability)
=X
i∈N
P(U≤1/i!) P(X=i) (because Uand Xare independent)
=X
i∈N
pqi−1
i!
=p
qX
i∈N0
qi
i!−1=p
q(eq−1).
3. Suppose that T, the total number of people who visit a store on a given day, is Poisson with
parameter λ(i.e., P(T=i) = e−λλi/i! for i∈N0). Further suppose that the sex of an individual
who visits the store is male with probability p, female with probability q= 1 −p, and independent
of the total number of visitors and sex of all other visitors. Finally, let Mdenote the number of
1/3
