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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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- Probability theory, continuous random variable, exponential random variable, conditional pmf, joint pdf fX,Y