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Unformatted text preview: Assignment 7 1. Let X, Y, Z have joint pdf f X,Y,Z ( x, y, z ) = k ( x + y + z ) for ≤ x ≤ 1 , ≤ y ≤ 1 , ≤ z ≤ 1 . (a) Find k . (b) Find f X ( x | y, z ) and f Z ( z | x, y ). (c) Find f X ( x ) , f Y ( y ), and f Z ( z ). 2. Show that f X,Y,Z ( x, y, z ) = f Z ( z | x, y ) f Y ( y | x ) f X ( x ). 3. Let U 1 , U 2 and U 3 be independent random variables and let X = U 1 , Y = U 1 + U 2 , and Z = U 1 + U 2 + U 3 . (a) Use the result in Problem 2 to find the joint pdf of X, Y , and Z . (b) Let the U i be independent uniform random variables in the interval [0,1]. Find the marginal pdf of Y and Z . Find the marginal pdf of Z . (c) Let the U i be independent zero-mean, unit variance Gaussian random vari- ables. Find the marginal pdf of Y and Z . Find the marginal pdf of Z . 4. A random experiment has four possible outcomes. Suppose that the experiment is repeated n independent times and let X k be the number of times outcome k occurs....
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- Winter '10
- Normal Distribution, Probability theory, joint characteristic function