hw2 - X ’s 6 Derive the moment generating function for...

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APPM 4/5520 Problem Set Two (Due Wednesday, September 7th) 1. Suppose that X 1 , X 2 iid exp ( rate = λ ). Find the distribution of X 1 X 1 + X 2 . (Name it!) 2. Let X 1 and X 2 have the joint pdf f X 1 ,X 2 ( x 1 , x 2 ) = 2 e - x 1 - x 2 I (0 ,x 2 ) ( x 1 ) I (0 , ) ( x 2 ) . (a) Find the marginal pdfs for X 1 and X 2 . (b) Suppose that Y 1 = 2 X 1 and Y 2 = X 2 - X 1 . Show that Y 1 and Y 2 are independent. 3. Suppose that X 1 , X 2 , . . . , X n is a random sample from the exponential distribution with rate λ . Find the pdf of Y = max( X 1 , X 2 , . . . , X n ). 4. Suppose that X 1 , X 2 , . . . , X n is a random sample from the uniform distribution over the interval (0 , 1). (a) Find the distribution of min( X 1 , X 2 , . . . , X n ). (Name it!) (b) Find the distribution of max( X 1 , X 2 , . . . , X n ). (Name it!) 5. Let X 1 , X 2 , . . . , X n be a random sample from the exponential distribution with rate λ . Find the expected value of nX (1) , where X (1) is the minimum of the
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Unformatted text preview: X ’s. 6. Derive the moment generating function for the exponential distribution with rate λ . Be sure to include an explantion as to why we need t < λ . 7. Required for 5520 students only Let X be a continuous random variable with pdf f X ( x ) and let Y = g ( X ) for some continuous invertible function g . Show that E [ g ( X )] = i ∞-∞ g ( x ) f X ( x ) dx. (Be sure to show the details for the limits of integration!) 8. Required for 5520 students only Let U 1 and U 2 be independent unif (0 , 1) random vari-ables. Show that X 1 and X 2 de±ned as X 1 = √-2 ln U 1 cos(2 πU 2 ) X 2 = √-2 ln U 1 sin(2 πU 2 ) are independent standard normal random variables....
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