PSTAT 109: Quiz 3
Instructions: Please show all work in an organized fashion for full credit.
Problem 1. Let = f1; 2; 3g, and consider the family of subsets A of :
A = f;;; f1; 2g; f2; 3g; f2g; f1; 3gg:
Explain why A is not a sigma-algebra.
Problem 2. If A and B are subsets of sample space , show that
P(A \ B) P(A) P(A [ B) P(A) + P(B):
Problem 3. A psychologist determined that the number of sessions required to obtain the
trust of a new patient is either 1, 2, or 3. Let X be a random variable indicating the number
of sessions required to gain the patients’s trust. The following probability distribution has
for x = 1; 2; or 3.
a. Write the probability distribution in the form of a table, similar to the ones found on
Lecture 5. Check if this function f is indeed a probability distribution.
b. What is the probability that it takes exactly 2 sessions to gain the patient’s trust?
c. What is the probability that it takes at least 2 sessions to gain the patient’s trust?
d. Calculate the expected value E(X) and variance V ar(X).
e. Suppose we dene a new random variable Y as
Y = 1[X=1] =
1; if X = 1;
0; if X 6= 1:
Write the probability distribution of Y in the form of a table, and interpret the meaning
of this random variable.
f. Calculate E(Y ) and V ar(Y ).