Problem 1.Let =f1;2;3g, and consider the family of subsets A of :

Instructions: Please show all work in an organized fashion for full credit.

Problem 1.Let =f1;2;3g, and consider the family of subsetsAof :

A=f;;;f1;2g;f2;3g;f2g;f1;3gg:

Explain whyAisnota sigma-algebra.

Problem 2.IfAandBare 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. LetXbe 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

;

forx= 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 functionfis indeed a probability distribution.

b. What is the probability that it takesexactly2 sessions to gain the patient’s trust?

c. What is the probability that it takesat least2 sessions to gain the patient’s trust?

d. Calculate the expected valueE(X) and varianceV ar(X).

e. Suppose we dene a new random variableYas

Y= 1[X=1]=

(

1;ifX= 1;

0;ifX6= 1:

Write the probability distribution ofYin the form of a table, and interpret the meaning

of this random variable.

f. CalculateE(Y) andV ar(Y).