# hw9-1 - a uniform random variable in the interval(1 3 Find...

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ORIE 3500/5500 – Engineering Probability and Statistics II Fall 2010 Assignment 9 Problem 1 Let X be a standard exponential random variable, and let Y = ± X if X 1 1 /X if X > 1 . Compute the cdf of Y . Problem 2 Let X be a continuous random variable with a density f X ( x ) = ± 1 + x if - 1 < x < 0 1 - x if 0 < x < 1 and zero otherwise. Find directly the density of Y = ± X if X > 0 - X 2 / 4 if X < 0 . Problem 3 The two most common types of errors made by program- mers are syntax errors and errors in logic. For a simple language the number of such errors is usually small. Let X denote the number of syntax errors and Y the number of errors in logic made on the ﬁrst run of a program. The joint pmf of X and Y is given by x i /y j 0 1 2 3 0 .400 .100 .020 .005 1 .280 .040 .010 .004 2 .040 .030 .009 .003 3 .009 .008 .007 .003 4 .008 .006 .006 .002 5 .005 .002 .002 .001 Find the pmf of the total number of errors Z = X + Y . Problem 4 A random vector ( X,Y ) has a joint pdf f X,Y ( x,y ) = ± 3 2 (2 x + y ) if 0 < x < y < 1 0 otherwise . ( a ) Find the joint pdf of T = XY and U = X/Y . ( b ) Find the marginal pdf of T . 1

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Problem 5 Let X and Y be independent random variables, X a standard uniform, and Y
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Unformatted text preview: a uniform random variable in the interval (1 , 3). Find the density of the product Z = XY . Caution : two diﬀerent cases come up here. Problem 6 Let U 1 and U 2 be independent standard uniform random variables. Find the density of Y = U 1-U 2 . Problem 7 A random vector ( X,Y ) has a joint pdf f X,Y ( x,y ) = ± 3 2 (2 x + y ) if 0 < x < y < 1 otherwise . ( a ) Find the conditional density of X given Y = y . ( b ) Find the conditional density of Y given X = x . ( c ) Calculate the conditional mean and the conditional variance of Y given X = x . Problem 8 For the data given in Problem 3: ( a ) ﬁnd the conditional pmf of X given Y = y j for all possible values y j ; ( b ) ﬁnd the conditional pmf of Y given X = x i for all possible values x i ; ( c ) calculate the conditional mean and the conditional variance of Y given X = x i for all possible values x i . Due: November 15, at 4 p.m. 2...
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hw9-1 - a uniform random variable in the interval(1 3 Find...

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