Conditional Expectation, change of variables, change of variables, CharacteristicsFcnby kliwin Tags: change of variables, characteristic, conditional, moment generating 

#1
Nov2112, 01:10 AM

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1.
Let T = (X,Y,Z) be a Gaussian for which X,Y,Z for which X, Y, Z are standard normals, such that E[XY] = E[YZ] = E[XZ] = 1/2. A) Calculate the characteristics function Φ_T(u,v,w) of T. B) Calculate the density of T. 2. Let X and Y be N(0,1) (standard normals), not necessarily independent. Calculate E[Max(X,Y)] using two different ways: A) Use the joint density of X and Y and use the fact that for two numbers x and y, max(x,y) = x if x > y and y if y > x. B) Use the change of variables x = x and y = ρx = u*sqrt(1ρ^2) in your integral. 3. Given random variables X and Y whose second moments exit, prove the triangle inequality E[(X+Y)^2]^(1/2) <= E[X^2]^(1/2) + E[Y^2]^(1/2) Help guys, this is for my Advanced Probability Course and I am stuck on it. :/ 


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