Question

2 Expectation, Co-variance and Independence [25pts + 5pts] Suppose X, Y and Z are three different random variables. Let X obeys Bernouli Distribution. The probability disbribution function is s 0.5 x=c 0.5 x= -c. c is a constant here. Let Y obeys the standard Normal (Gaussian) distribution, which can be written as Y ~ N(0,1). X and Y are independent. Meanwhile, let Z = XY p(x) = { 0.5 • Are Y and Z independent? (Just clarify) [3pts] • Show your conclusion for the above question with an specific example. (Bouns) [5pts]

Answer 1

No, Y and Z are not independent because from formula Z = XY ; it is clear that if Y=0; then Z can not take any value other than 0.

Example : Let Y= 0; Then the only value Z takes is 0. So, knowing value of Y, does give us lot of information on Z hence they must be dependent. For eg. P( Z = 0 ) = P(Y=0) < 1 ; However, P(Z=0 | Y=0) = 1 ,so we see ;

P(Z=0) is different from P(Z=0 | Y=0) .  

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