Sample problems for the second midterm. 3 * second midterm exam apr. 281 1 2, 3, 6, 10, 17 Apr. 10 * Please do the additional exercises:.* evernote Consider the coupon collecting problem with three different types of coupons. And let T be the number of boxes needed until you first possess all three types. Find P( tk e( T ) and Var( T ) using a probability generating function. B.* Flip a fair coin repeatedly until you get two consecutive heads.

Find cov( x, y ) and ρ(X,Y). 226 31 32, 33, 35, 36 39 Mar. 27 * Please do the additional exercise:.* N people arrive separately to a professional dinner. Upon arrival, each person looks to see if he or she has any friends among those present. That person then either sits at the table of a friend or at an unoccupied table if none of those present is a friend. Assuming that each of the nc2 pairs of people are, independently, friends with probability p, find the expected number of occupied tables. Cf Ross, Ch 7, Prob.

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B.* Roll five studies dice. Let X be the smallest number of the five dice. Find P (X x fX(x e (X). C.* Let x poisson(λ). Find fX( x x is odd ) and E ( x x is odd ).

226 1-3, 6, 9 10, 12, 14, 15, 18, 21, 24, 25* Mar. 13 * Please do the additional exercise:.* Two cards are selected at random without replacement from a standard deck. Let X be the number of kings and Y be the number of clubs. Determine the joint pmf f(x,y). Find the marginal pmfs. Are x and y independent?

Find P( x is odd P( x is even P( x k p( 2 x 9 x 4). For 1 k n, find P( x k x n ). Letting k be a natural number and y g(X) where g(x) min (x,k), find pmf fY(y). Find E( 1/X ). B.* An unfair coin is tossed repeatedly.

Suppose that the events of getting a head on the i th toss are independent and P( H ). Let X be the number of tosses it takes to first get three heads. Derive the formulas for E( X ) and Var( X ). 151 14 30, 33, 35, 40, 42 45, mar. 6 * Please do the additional exercises:. Let X be a random variable with finite second moment. Let μ e (X) and σ2 var (X). Let y (x -. Show E (Y)0 and Var (Y)1.

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11 *., 11, 16, feb. Please do the additional exercises:.* Problem 1 with 13 sound, 3 rotten and choose 4 randomly. A.* Roll four dice. X is the number of sixes. Find twist d, fx, fx, e(X). B.* Let D1,2,3,.,n and fX(i)ci for i. Find c to make fx. 151 18, 25 27, 29, feb. 27 * Please do the additional exercises:.* Suppose x geom(p).

15 19,. 76 1 3, 5 6 - 8,. 76 11, 12 15, 19 20, 23 27, 34, 35, feb., 2 4,., 15, 17, 19, feb. 13, sample problems for the first midterm. first midterm exam, feb.

By david Stirzaker, cambridge University Press, 2003. You are responsible for all of william the listed problems. Please hand in only the starred problems. Please make sure your papers are self-contained. Copy or paraphrase the statement of the problem. Use English sentences to give at least a minimal explanation of your answer. State any theorems or formulas you use. Atk students homework is due the wednesday following the Friday due date for regular students.

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