# assn5 - f ( x ) = 12 x 2 (1-x ) , < x < 1, zero...

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Math 361, Problem set 5 Due 10/04/10 1. (1.6.8) Let X have the pmf p ( x )=( 1 2 ) x , x =1 , 2 , 3 , . . . , and zero else- where. Find the pmf of Y = X 3 . 2. (a) Pick a card from a standard deck. Let X denote the rank of the card(counting ace as one, J=11, Q=12, K=13.) Find the pmf of X . (b) Pick two cards from a deck, with replacement. Let Y denote the highest rank picked. Find the pmf of Y . 3. (1.7.8) A mode of a distribution of one random variable X is a value of x that maximizes the pdf or pmf. For X of the continuous type, f ( x ) must be continuous. If there is only one such x , it is called the mode of the distribution. Find the mode of each of the following distributions: (a) p ( x ) = ( 1 2 ) x , x, 1 , 2 , 3 , . . . , zero elsewhere. (b)
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Unformatted text preview: f ( x ) = 12 x 2 (1-x ) , < x < 1, zero elsewhere. (c) f ( x ) = 1 2 x 2 e-x , < x < ∞ , zero elsewhere. 4. (1.7.14) Let X have the pdf f ( x ) = 2 x , 0 < x < 1, zero elsewhere. Compute the probability that X is at least 3 4 given that X is at least 1 2 . 5. (1.7.17) Divide a line segment into two parts by selecting a point at ran-dom. Find teh probability that the larger segment is at least 3 times the shorter. Assume the point is chosen uniformly. 6. (1.7.22) Let X have the uniform pdf f X ( x ) = 1 π for-π 2 < x < π 2 . Find the pdf of Y = tan( X ). This is the pdf of a Cauchy distribution . 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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