100AHW7 - STAT 100A HWVII Due next Fri Problem 1: Suppose Z...

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STAT 100A HWVII Due next Fri Problem 1: Suppose Z N(0 , 1). The density of z is f ( z ) = 1 2 π e - z 2 / 2 . E[ Z ] = 0, Var[ Z ] = 1. Let X = μ + σZ , where σ > 0. (1) Find the probability density function of X . (2) Calculate E[ X ] and Var[ X ]. Problem 2: Suppose U Uniform(0 , 1). Let T = - log U/λ . (1) For t > 0, calculate P ( T > t ). (2) Find the probability density function of T . Problem 3: 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 4: For two continuous random variables X and Y with a joint density function f ( x,y ), prove (1) E[ X + Y
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