Stat 432: Homework 4
1. Suppose that X and Y are independent Poisson random variables with respective means of
 x and  y . What is the conditional probability function of X given that X+Y = n?
2. The number of telephone calls coming into a service center during a one hour period is a
Poisson random variable with an average rate of 5. Of the calls coming into the center the
service representative can satisfy the caller 80% of the time. Assume that the ability to
satisfy one caller is independent of the ability to satisfy another caller and that the 80%
satisfaction is the same for all callers.
a. What is the probability that exactly 2 callers will be satisfied during a one hour period?
b. What is the probability that at least 1 caller will be satisfied during a one hour period?
c. What is the mean (expected) number of satisfied customers during a one hour period?
d. What is the probability that all of the callers during a one hour period will be satisfied?
3. An exam has 25 multiple choice questions on it. A student decides to answer only those
questions that she thinks she can answer correctly. Assume that she has studied only 80% of
the material so the chance that she chooses to answer a question is 0.80. Also assume that
choosing to answer one question is independent of choosing to answer another question.
Given that she chooses to answer a question, assume that the chance she answers it correctly
is 0.95 (she studied really hard the material that she studied). Also assume that answering
one chosen question correctly is independent of answering other chosen questions correctly.
a. What is the probability she answers all 25 questions correctly?
b. What is the chance she answers 20 or more questions correctly?
c. What is the mean (expected) number of correctly answered questions?
d. Another student studied 95% of the material but didn’t study that material as well so only
has an 80% chance of answering a chosen question correctly. How do the answers for a),
b) and c) change for this student?
4. The Smith family has 4 children. The Jones family has 6 children. Assume that the
probability that a child is a boy is the same as the probability that a child is a girl (p = (1 – p)
= 0.5). Also assume that whether a child is a boy or girl is independent with each family and
the two families are also independent in terms of whether a child is a boy or a girl. Let N be
the number of female children in both families. If N = 4, what is the probability that the
Smith family has exactly 2 boys?
5. Problem 2.1.10 on page 52 of the text.
Extra Credit: Assume that the probability that a child is a girl is p and the probability that a child
is a boy is 1 – p. Assume that whether a child is a boy or a girl is independent of the other
children’s gender. A family decides to continue to have children until they have their first boy.
Let N be the number of girls in the family. Assume that each girl in the family will go to college
with probability q and not go to college with probability 1 – q. Assume that whether a girl goes
to college or not is independent of what the other girls do. Find the marginal distribution of Y the
number of girls in the family who go to college. Hint: Calculate the probability that Y = 0
separately.
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