Probability Let random variables X1 and Y be jointly distributed with distribution p(x,y). You can assume that they are

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Probability Let Random Variables X1 And Y Be Jointly Distributed With Distribution P X Y You Can Assume That They Are 1 (86.94 KiB) Viewed 75 times

Probability Let random variables X1 and Y be jointly distributed with distribution p(x,y). You can assume that they are jointly discrete so that p(x,y) is the probability mass function (pmf). Show the following results by using the fundamental properties of probability and random variables. = (a) E[X] E[E[X|Y]], where E[X] = {zxp(x) denotes statistical expectation of X and E[X|Y] = {x xp(|Y) denotes conditional expectation of X given Y. (b) E[I[X E C]] = P(X EC), where I[X E C) is the indicator function of an arbitrary set C (i.e. I[X E C] = 1 if X EC and 0 otherwise. (c) If X and Y are independent then E[XY] = E[X]E[Y] (d) If X and Y take values in {0, 1} and E[XY] = E[X]E[Y], then X and Y are independent.