Posted: November 21st, 2022

PSTAT 109: Quiz 3

Winter 2014

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

been proposed.

f(x) =

x

6

;

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 ).

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