Pstat 109: quiz 3 winter 2014

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) =






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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