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PSTAT 109: Quiz 3 Winter 2014




Question;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 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).


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