HW06 - F if for every x , lim n P ( n Y n x ) = F ( x ) ....

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STAT 36-217, HW 6, due Thursday 10/15/2009, 10:30 AM PLEAE USE THIS AS THE COVER PAGE Your Name: 1
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1. What is the value of Γ( n + 1 / 2)? What is the value of Γ( n )? Use these to find Γ(6 . 5) and Γ(11). 2. Suppose X is distributed as exponential with parameter λ = 2. Find E [ X 3 ] , E [ X 4 ] , E [ X 5 ] . 3. Suppose X N (0 , 4). Find E [ X 3 ] , E [ X 4 ] , E [ X 6 ] . 4. If X is normally distributed with mean 1 and variance 4. Use the normal table to find P (2 < X < 4). 5. Flip a coin with 200 times independently. Suppose that for each flip, the chance of landing head is 0 . 4. Find the probability that we have 75 or less heads. Suppose that for each flip, the chance of landing head is 0 . 01. Find the proba- bility that we have 3 head or more heads. 6. Given n independent uniform random variables X i U (0 , 1), 1 i n . Let Y n = min { X 1 ,X 2 ,...,X n ) . Show that P ( n · Y n x ) = (1 - x n ) n for any x (0 , 1). What is lim n →∞ P ( n · Y n x )? We say that n · Y n weakly converges to cumulative distribution
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Unformatted text preview: F if for every x , lim n P ( n Y n x ) = F ( x ) . Find the cumulative distribution F to which n Y n converges weakly. What is the name of this distribution? 7. The joint density of X and Y is given by f ( x,y ) = e-y y , &lt; x &lt; y, &lt; y &lt; . Verify that this is a joint density. Find E [ X 2 | Y = y ]. 8. The joint density of X and Y is given by f ( x,y ) = e-x/y e-y y , &lt; x &lt; , &lt; y &lt; . Verify that this is a joint density. Find marginal density of Y . Show E [ X | Y = y ] = y . 9. Let X be exponential distribution with parameter . What is the distribution of ( X | X &gt; 1)? Find E [ X | X &gt; 1]. 10. Let X be uniform over (0 , 1). What is the distribution of ( X | X &lt; 1 / 2)? Find E [ X | X &lt; 1 / 2]. 2...
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This note was uploaded on 09/30/2010 for the course STATISTICS 36-217 taught by Professor Jin during the Fall '09 term at Carnegie Mellon.

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HW06 - F if for every x , lim n P ( n Y n x ) = F ( x ) ....

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