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Unformatted text preview: P ( Y 5). Then, nd its ex-act value using properties of Poisson random variables. How do the answers compare? 1 5. Suppose that X = X 1 + + X k , where the X i s are all independent and have the same distribution. Find a formula for E ( X 4 ) in terms of the moments 1 = E ( X 1 ), 2 = E ( X 2 1 ) , 3 = E ( X 3 1 ) and 4 = E ( X 4 1 ). (Hint: This is almost a triviality if you use moment generating functions.) 6. Let T denote the triangle with corners at (0 , 0) , (0 , 1), and (1 , 1), and dene the function f ( x, y ) = b x + cx 2 y 2 , if ( x, y ) T ; , otherwise . Find the constant c that makes this function into a probability density function. 7. Suppose X and Y are i.i.d. random variables having pdf f ( t ) = b 1000 /t 2 , if t > 1000; , otherwise . Suppose Z = X/Y . Determine the pdf for Z . 2...
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