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2010100AHW8

# 2010100AHW8 - STAT 100A HWVIII Problem 1 Consider the...

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STAT 100A HWVIII Problem 1: Consider the following joint probability mass function p ( x,y ) of the discrete random variables ( X,Y ): x \ y 1 2 3 1 .1 .1 .1 2 .2 .1 .2 3 .1 .05 .05 (1) Calculate p X ( x ) for x = 1 , 2 , 3. Calculate p Y ( y ) for y = 1 , 2 , 3. (2) Calculate P ( X = x | Y = y ), and calculate P ( Y = y | X = x ), for all pairs of ( x,y ). (3) Calculate E[ X ] and E[ Y ]. Calculate Var[ X ] and Var[ Y ]. (4) Calculate E[ XY ]. Calculate Cov( X,Y ). Calculate Corr( X,Y ). Problem 2: For two continuous random variables X and Y with a joint density function f ( x,y ), prove (1) E[ X + Y ] = E[ X ] + E[ Y ]. (2) Var[ X + Y ] = Var[ X ] + Var[ Y ] + 2Cov( X,Y ). (3) If X and Y are independent, i.e., f ( x,y ) = f X ( x ) f Y ( y ), then Cov( X,Y ) = 0. Explain that the reverse may not be true, i.e., if Cov( X,Y ) = 0, X and Y may not be independent, by giving a counter example. Problem 3:

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2010100AHW8 - STAT 100A HWVIII Problem 1 Consider the...

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