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Unformatted text preview: Stat 116 Homework 6 Due Wednesday, May 21st. Please show work and justify answers. No credit for a final answer with no explanation, even if the answer is correct. 1. Let X and Y be distributed std. Normal with correlation between X and Y equal to ρ . Compute the joint and marginal distributions of X + Y and X Y , are X + Y and X Y independent? 2. Let U ∼ χ 2 ( r ) and V ∼ χ 2 ( s ) be independent with s > 2. (a) Compute E parenleftBig U/r V/s parenrightBig . (b) Use part (a) to compute E t 2 where t has the student’s t distribu tion on s degrees of freedom. ( hint : The r.v. U/r V/s from part (a) is known as the F distribution on r and s degrees of freedom how does this r.v. relate to the t .) 3. Let X i , i = 1 ,...,n be independent r.v.’s with mean μ and variance σ 2 . Show E ∑ n i =1 ( X i ¯ X ) 2 = ( n 1) σ 2 , where ¯ X = 1 n ∑ n i =1 X i is the sample mean. This result justifies using 1 n 1 ∑ n i =1 ( X i ¯ X ) 2 as our sample variance, i.e. we divide bysample variance, i....
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 Spring '07
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 Correlation, Normal Distribution, Probability, Variance, Triangular matrix, X1 =L Z2

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