307_assignment_112208

307_assignment_112208 - Problem Assignment Math 307/607 due...

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Unformatted text preview: Problem Assignment Math 307/607 due Thursday December 4 1. Pedestrian Review: the random variables X , Y have joint density of the form f X ,Y ( x, y ) = c( x + y ), 0 < x, 0 < y, x + 2 y < 2, where c is an unspecified constant. a) Determine c. b) Find the marginal densities f X , fY . c) Find μ X , μY , σ X , σ Y . d) Find the covariance Cov ( X , Y ) . e) Find the correlation ρ ( X , Y ) . f) Find the conditional expectation E ( X | Y = .5) . NOTE: you should be able to do a)‐f) without conceptual difficulty. They may be tedious, but you should not be stumped at all. 2. Suppose we have random variables X , Y with μ X = 5, μY = 3, σ X = 2, σ Y = 3, and correlation ρ ( X , Y ) = −.6. Find the mean and standard deviation of W = X + 2Y + 1. 3. If the random variable Z has the normal distribution with mean 0, standard deviation 1, then the moment generating function mZ ( t ) = exp t 2 2 . Taking this as known, use this to find 2 3 4 2 () E ( Z ) , E ( Z ) , E ( Z ) , E ( Z ) . Find Var ( Z ) . 4. If the random variable X has the Poisson distribution with mean λ > 0, then the moment generating function of X is mX ( t ) = exp λ et − 1 . Use this fact to show: if V , W are independent Poisson random variables with means λV , λW respectively, then V + W is Poisson with mean λV + λW . 5. Suppose X 1 , X 2 ,… , X 1000 are independent random variables, each uniformly distributed on the unit interval [ 0,1] . Let S = 1000 i =1 (( )) ∑ X . i (a) Find E ( S ) and Var ( S ) . (b) Explain why S has an approximately normal distribution. (c) Find the middle 95% of likely outcomes of S. ...
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