strict due date.
MATH 464
HOMEWORK 3
SPRING 20
1
3
The following assignment is to be turned in on
Thursday, February 7, 2013.
1. Three couples are invited to a dinner party. They will independently
show up with probabilities 0.9, 0.8, and 0.75 respectively. Let N be the
number of couples that show up. Calculate the probability that N = 2
2. Statistics show that 5% of men are color blind and 0.25% of women are
color blind. If a person is randomly selected from a room with 35 men
and
65 women, what is the likelihood that they are color blind?
3. Do Exercise 26 on page 14 of the book.
4. On a multiple choice exam with four choices for each question, a student
either knows the answer to a question or marks it at random. Suppose the
student knows the answers to 60% of the exam questions. If he marks the
answer to question 1 correctly, what is the probability that he knows the
answer to that question?
5. In a certain city, 30% of the people are conservative, 50% are liberals, and
20% are independents. 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 this person is a liberal?
6. Let (Ω,F, P ) be a probability space and suppose that {An}∞n=1 is an
increasing sequence of events. For each integer n ≥ 1, set
Cn =
{
A1 if n = 1
An \An−1 for n ≥ 2.
Show that the Cn’s are mutually disjoint and that
∞⋃
n=1
An =
∞⋃
n=1
Cn .
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2 SPRING 2013
7. Let (Ω,F, P ) be a probability space and suppose that {An}∞n=1 is a
sequence of events. Set
Bn =
∞⋃
m=n
Am and Cn =
∞⋂
m=n
Am
It is clear that Bn is a decreasing sequence of events, while Cn is an increasing
sequence of events. Show that
B =
∞⋂
n=1
Bn = {ω ∈ Ω : ω ∈ An for infinitely many values of n}
and
C =
∞⋃
n=1
Cn = {ω ∈ Ω : ω ∈ An for all but finitely many values of n}
8. Do exercise 4 on page 24 of the book.
9. Suppose we roll two fair 6-sided dice. Let X be a random variable
corresponding to the minimum value of the two rolls. Find the probability
mass function fX corresponding to the random variable as a table of values
(see below).
10. The probability mass function of a discrete random variable X is given
below as a table of values. Compute the following:
a) the probability that X is even (here we regard 0 and -4 as even)
b) the probability that 1 ≤ X ≤ 8
c) the probability that X is -4 given that X ≤ 0
d) the probability that X ≥ 3 given that X > 0
x -4 -1 0 2 4 5 6
fX(x) 0.15 0.2 0.1 0.1 0.2 0.2 0.05
MATH 464
HOMEWORK 4
SPRING 20
1
3
The following assignment is to be turned in on
Thursday, February 14, 2013.
1. Let X be a discrete random variable on a probability space (Ω,F,P).
Let g : R → R be a function and set Y = g(X), i.e. Y : Ω → R is defined by
Y (ω) = g(X(ω)) for all ω ∈ Ω .
Prove that Y is a discrete random variable.
2. Let 0 < p ≤ 1 and consider a function X with range {1, 2, 3, · · ·} and corresponding numbers
P(X = k) = p(1 −p)k−1 for any integer k ≥ 1 .
Prove that X is a discrete random variable by showing that the sum of
the above probabilities is 1. This is the geometric random variable with
parameter p.
3. Let X be a Poisson random variable with parameter λ > 0. Compute
the following:
a) P(2 ≤ X ≤ 4)
b) P(X ≥ 5)
c) P(X is even)
give each answer in exact form and, with the choice of λ = 2, give a decimal
approximation to the above which is accurate to 3 decimal places.
4. Let X be a discrete random variable whose range is {0, 1, 2, 3, · · ·}. Prove
that
E(X) =
∞∑
k=0
P(X > k) .
5. Compute the expected value of the geometric random variable with pa-
rameter 0 < p ≤ 1. Hint: Use problem 4 above.
6. Let X be a binomial random variable with parameters 0 ≤ p ≤ 1 and
n > 0 an integer. For any 0 ≤ k ≤ n, denote by Pk = P(X = k). Compute
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2 SPRING 2013
the ratio
Pk−1
Pk
for 1 ≤ k ≤ n.
Show that this ratio is less than one if and only if k < np + p. This shows that the most probable values of X are those near np.
7. Let X be a Poisson random variable with parameter λ > 0. Let g : R → R
be the function g(x) = x(x− 1). Set Y = g(X). Find E(Y ).
8. Let X be a function whose range is {1, 2, 3, · · ·}. Consider the values
P(X = n) =
1
n(n + 1)
for any n ≥ 1 .
Does this function X define a discrete random variable? If so, what is E(X)?