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) =

 

 

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 ).