Texas A&M University
Department of Mathematics
Volodymyr Nekrashevych
Summer 2014
Math 411 — Problem Set 3
Issued: 06.23
Due: 06.30
3.1. A, B, and C are events with P (A) = 0.3, P (B) = 0.4, and P (C) = 0.5,
A and B are disjoint, A and C are independent, and P (B|C) = 0.1.
Find P (A ∪ B ∪ C).
3.2. John takes the bus with probability 0.3 and the subway with probability 0.7. He is late 40% of the time when he takes the bus, but only
30% of the time when he takes the subway. What is the probability
that he is late for work?
3.3. Two boys, Charlie and Doug, take turns rolling two dice with Charlie
going first. If Charline rolls a 6 before Doug rolls a 7 he wins. What is
the probability that Charlie wins?
3.4. In a certain city, 30% of the people are Conservatives, 50% are Liberals,
and 20% are independent. In a given election, 2/3 of the Conservatives
voted, 80% of the liberals voted, and 50% of the Independents voted.
If we pick a voter at random what is the probability that he or she is
Liberal?
3.5. Suppose X has density function 6x(1 − x) for 0 < x < 1 and 0
otherwise. Find (a) EX, (b) E(X 2 ), and (c) var(X).
3.6. Suppose X has density function 4x3 for 0 < x < 1 and 0 otherwise.
Find (a) the distribution function, (b) P (X < 1/2), (c) P (1/3 < X <
2/3), (d) the median.
3.7. Suppose X has density function f (x) for a ≤ x ≤ b and Y = cX + d,
where c > 0. Find the density function of Y .
3.8. Suppose X and Y have joint density f (x, y) = 1 for 0 < x, y < 1. Find
P (XY ≤ z).
3.9. Suppose X is uniform on (0, 1) and Y = X. Find the joint distribution
function of X and Y .