UCSB PSTAT 109 Statistics for Economics Question-Quiz 3 Solution

Problem 1

 Let Ω = {1, 2, 3}, and consider the family of subsets A of Ω:

A = {∅, Ω, {1, 2}, {2, 3}, {2}, {1, 3}}.

Explain why A is not a sigma-algebra.

Problem 2

 If A and B are subsets of sample space Ω, 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’ trust. The following probability distribution has

been proposed.

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 define a new random variable Y as

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 )