assn4 - X denote the number of heads ²ipped. Then the pmf...

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Math 361, Problem set 4 Due 9/20/10 1. (1.4.26) Person A tosses a coin and then person B rolls a die. This is re- peated independently until a head or one of the numbers 1 , 2 , 3 , 4 appears, at which time the game is stopped. Person A wins with the head, and B wins with one of the numbers 1 , 2 , 3 , 4. Compute the probability A wins the game. 2. (1.5.6) Let the probability set function P X ( D ) of the random variable X be P X ( D )= ± D f ( x ) dx , where f ( x )= 2 x 9 , for x ∈ D = { x :0 < x < 3 } . Let D 1 = { x :0 < x < 1 } , D 2 = { x :2 < x < 3 } . Compute P X ( D 1 )= P ( X D 1 ), P X ( D 2 )= P ( X D 2 ) and P X ( D 1 D 2 )= P ( X D 1 D 2 ) . 3. (1.5.5) Let us select Fve cards at random and without replacement from an ordinary deck of playing cards. (a) ±ind the pmf of X , the number of hears in the hand., (b) Determine P ( X 1). 4. A weighted coin, with head probability 1 10 is ²ipped n times, where n is divisible by 10. Let
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Unformatted text preview: X denote the number of heads ²ipped. Then the pmf of X is p ( k ) = ( n k ) (1 / 10) k (9 / 10) n-k . Show which value of k this maximizes this. Hint: Look at the ratio: p ( k ) /p ( k + 1). When k is small, this is less than one, when k is large, this is bigger than one. ±ind the value of k when p ( k ) /p ( k + 1) ≈ 1. Why does this work? 5. (1.6.3) Cast a die a number of independent times until a six appears on the up side of the die. (a) ±ind the pmf p ( x ) of X , the number of casts needed to obtain that Frst six. (b) Show that ∑ ∞ x =1 p ( x ) = 1. (c) Determine P ( X = 1 , 3 , 5 , 7 ,... ) . (d) ±ind the cdf F X ( x ) = P ( X ≤ x ). 1...
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This note was uploaded on 10/26/2010 for the course MATHCS Math 316 taught by Professor Dr.paulhorn during the Fall '10 term at Emory.

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