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prob1-01-12-09-p1

prob1-01-12-09-p1 - MATH 11300 Problem Sheet 9 Probability...

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MATH 11300 Probability AUTUMN 2009 Problem Sheet 9 Questions marked * are to be handed in. *1. Let X be a continuous random variable taking values in the interval [ - 3 , 3], with probability density function (pdf) f X given by f X ( x ) = (9 - x 2 ) / 36 - 3 x 3 0 otherwise . (a) Find the distribution function F X for X . (b) Find P ( X < 0) , P ( - 2 X 1) and P ( X > 2 . 5). 2. Let X be a continuous random variable with distribution function F X given by F X ( x ) = 1 - (1 + x ) e - x x > 0 0 otherwise . Find the density function f X for X . Hence find E ( X ) and Var( X ) *3. Let Y be a continuous random variable taking values in the interval [0 , 1]. Assume Y has probability density function (pdf) proportional to y (1 - y ), i.e. there exists a fixed constant c such that the pdf f Y is given by f Y ( y ) = cy (1 - y ) 0 y 1 0 otherwise. (a) Find the value c must take for this to be a pdf. (b) Sketch the pdf f Y . (c) Find and sketch the distribution function F Y for Y . *4. Let Y = X 2 where X U ( - 1 , 2). Find the distribution function F Y for Y . Hence find the density function for Y . [Hint: For F Y ( y
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