UNIVERSITY OF CALIFORNIA, BERKELEY

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1 UNIVERSITY OF CALIFORNIA, BERKELEY DEPARTMENT OF STATISTICS STAT 34: Concepts of Probability Spring 24 Instructor: Antar Bandyopadhyay Solution to the Midterm Examination. A point X, Y is randomly selected from the following finite set of points on the plane { } x, y y x n. a What are the marginal distributions of X and Y? Is the pair X, Y exchangeable? Explain your answer. [2 + ] First note that the total number of points in the given set is n n n +. 2 The values of the random variable X are {, 2,, n}. Fix x n, then P X x x 2x P X x, Y y n n +. The values of the random variable Y are {, 2,, n}. Fix y n, then P Y y y n P X x, Y y xy 2 n y +. n n + Note that always Y X and hence the pair X, Y is not exchangeable. b What is the distribution of the random variable X + Y? [5] Let Z : X + Y, then the values of Z are {2, 3,, 2n}. Fix 2 z 2n, then 2z minz,n z 2 nn+ if 2 z n + P Z z P X x, Y y x z 2 2n z + 2 nn+ if n + < z 2n, where u means the least integer greater or eual to the real number u.

2 c Find the conditional distribution of X given X + Y n. [4] Given [X + Y n] the values of the random variable X are { n 2, n 2 +,, n }. Fix n 2 x n, then P X x X + Y n P X x, Y n x P X + Y n This X X + Y n Uniform { n 2, n 2 +,, n }. n n Suppose N Negative-Binomial 2, p where < p <. Let X be a random variable such that the conditional distribution of X given N n is Uniform {, 2,, n }. Find the marginal distribution of X. What is the conditional distribution of N given X. [8 + 4] First we note that the values of X are {, 2, }. Now fix x, So we get X Geometric p. P X x nx+ nx+ P X x N n P N n n n p2 p n 2 p 2 p x p p p x. Now, give [X ], the values of N are {, 2, }. Fix n, P X N n P N n P N n X P X Thus N X + Geometric p. p p n. 3. In a Quidditch trail Harry ased Ginny to try to score a goal with Ron as the eeper. Ginny who is an excellent player has a chance of scoring a goal 9% of the times. The game was stopped as soon as Ginny scored a goal. It too exactly 9 minutes for her to score a goal. Find the expected time Ginny too for each of her tries. Explain the assumptions you are maing. [ + 2] We assume that each try of Ginny are independent Bernoulli trials with success probability.9. We also assume that that each try Ginny taes eual amount of time. Let X be the number of trials Ginny made, then X Geometric.9. So the expected time Ginny 2

3 too for each of her tries is [ ] 9 E X 9p 9p 9p 9p 9.9. [p.9 and.] t dt t dt dt t 9p log p 8 log Ron was given the tas of sending out the invitations for the wedding of Ginny and Harry. They wanted to invite of their friends. Unfortunately, Ron did not realized that his charm made the invitation letters to be randomly placed in the addressed envelop. The outcome was a disaster, it may have been that an invitation has gone inside an envelop addressed to a different person, or an envelop containing more than one invitations or none at all. In all such cases the recipients were either confused or upset and they did not show up. Only the invitees who received the invitations inside a correctly addressed envelop came for the wedding. Let X be the total number of people who came for the wedding. a Find P X and P X. [3 + 4] Let A i : i th invitation is the only one which has gone inside the i th envelop. Then by definition X Ai. Thus, Further, i P X P A i i. P X P X P A i i. The third euality is by the inclusion-exclusion formula. 3

4 b What is the expected number of people who came for the wedding? [5] This is given by 99 E [X] P A i i 5. There was a strange news reported in the seven o cloc news, apparently a newly built bridge in northern England collapsed during the pea commute hours without any problem reported about it earlier. The total number of casualties was yet to be determined but it was somewhere in between 6 and 8. Harry was clever to figure out that this must have been a wor of the Dar Lord and his followers. He uicly reported it to Professor Dumbledore, who told him that from his prior experiences, he new that the Dar Lord must had decided on the exact number of people he wanted to harm by either rolling a fair sided die and then taing the number which came on top, or tossing a coin times independently and then too the number of heads. He also did not thin that the Dar Lord had any preference on the choice of his method. When Harry told this to his friend Hermione, she was uic to mae a guess about the actual method the Dar Lord used. a What was Hermione s guess? Explain your answer. [6] Let Z be the exact number of casualties determined by the Dar Lord and let I is the indicator random variable taing value I if he rolled the sided die to determine Z, otherwise it taes the value I. Then the conditional distributions of Z I Uniform {, 2,, } and Z I Binomial, 2. Also P I P I 2. Now the event A : [6 Z 8] has been observed. So P A I P I P I A P A P A I P I P A I P I + P A I P I P 6 U 8 P 6 U 8 + P 6 V 8 where U Uniform {, 2,, } and V Binomial, 2. So P 6 U 8 2 and P 6 V 8 Φ6. Φ.9 Φ.9 Φ Note E [V ] 5 and Var V 25. Thus P I A So Hermione s must had guessed that the Dar Lord rolled the sided die. 4

5 b Will she mae a different guess if the news channel said the number of casualties were between 4 to 6? Explain your answer. [6] Let B : [4 Z 6], then similar argument will show P B I P I P I B P B P B I P I P B I P I + P B I P I P 4 U 6 P 4 U 6 + P 4 V 6 where U Uniform {, 2,, } and V Binomial, 2. Now P 4 U 6 2 and P 4 V 6 Φ2. Φ 2. 2Φ Once again note that E [V ] 5 and Var V 25. Finally, P I B So yes, in that case she must had guessed that the Dar Lord tossed the fair coin times. 5