Description
1 Safeway Monopoly Cards
It’s that time of the year again – Safeway is offering its Monopoly Card promotion. Each time you
visit Safeway, you are given one of n different Monopoly Cards with equal probability. You need
to collect them all to redeem the grand prize.
Let X be the number of visits you have to make before you can redeem the grand prize. Show that
var(X) = n
2
∑
n
i=1
i
−2
−E(X). [Hint: Does this remind you of a particular problem? What is the
expectation for this problem?]
2 Geometric Distribution
Two faulty machines, M1 and M2, are repeatedly run synchronously in parallel (i.e., both machines
execute one run, then both execute a second run, and so on). On each run, M1 fails with probability
p1 and M2 fails with probability p2, all failure events being independent. Let the random variables
X1, X2 denote the number of runs until the first failure of M1, M2 respectively; thus X1, X2 have
geometric distributions with parameters p1, p2 respectively. Let X denote the number of runs until
the first failure of either machine.
(a) Show that X also has a geometric distribution, with parameter p1 + p2 − p1 p2.
(b) Now, two technicians are hired to check on the machines every run. They decide to take turns
checking on the machines every hour. What is the probability that the first technician is the
first one to find a faulty machine?
CS 70, Fall 2018, HW 12
3 Geometric and Poisson
Let X be geometric with parameter p, Y be Poisson with parameter λ, and Z = max(X,Y). Assume
X and Y are independent. For each of the following parts, your final answers should not have
summations.
(a) Compute P(X > Y).
(b) Compute P(Z ≥ X).
(c) Compute P(Z ≤ Y).
4 Darts
Alvin is playing darts. His aim follows an exponential distribution; that is, the probability density
that the dart is x distance from the center is fX(x) = exp(−x). The board’s radius is 4 units.
(a) What is the probability the dart will stay within the board?
(b) Say you know Alvin made it on the board. What is the probability he is within 1 unit from the
center?
(c) If Alvin is within 1 unit from the center, he scores 4 points, if he is within 2 units, he scores
3, etc. In other words, Alvin scores b5−xc, where x is the distance from the center. What is
Alvin’s expected score after one throw?
5 Exponential Practice
(a) Let X1,X2 ∼ Exponential(λ) be independent, λ > 0. Calculate the density of Y := X1 + X2.
[Hint: One way to approach this problem would be to compute the CDF of Y and then differentiate the CDF.]
(b) Let t > 0. What is the density of X1, conditioned on X1 +X2 = t? [Hint: Once again, it may
be helpful to consider the CDF P(X1 ≤ x | X1 + X2 = t). To tackle the conditioning part, try
conditioning instead on the event {X1 +X2 ∈ [t,t +ε]}, where ε > 0 is small.]
6 Uniform Means
Let X1,X2,…,Xn be n independent and identically distributed uniform random variables on the
interval [0,1] (where n is a positive integer).
(a) Let Y = min{X1,X2,…,Xn}. Find E(Y). [Hint: Use the tail sum formula, which says the
expected value of a nonnegative random variable is E(X) = R ∞
0 P(X > x)dx. Note that we can
use the tail sum formula since Y ≥ 0.]
(b) Let Z = max{X1,X2,…,Xn}. Find E(Z). [Hint: Find the CDF.]
CS 70, Fall 2018, HW 12 2
