Consider a pair of discrete random variables (X, Y) which can take any of the fo

Question

Consider a pair of discrete random variables (X, Y) which can take any of the following values S = {(0,1),(2,1),(2,2),(4,0), (4,2), (5,2), (3,0)} with equal probability. Draw the joint PMF. Find the PMF of Y. Find mean of X. Which experimental value(s) x of the random variable X maximize E[Y|X = x], the conditional expected valued of Y given the experimental value of X? Which experimental value(s) y of the random variable Y maximize Var(X|Y), the conditional variance of X given the experimental value of F? Prepare a neat, fully labeled sketch of the PMF pR(r), where the random variable R is defined by the relation R = min[X,F] Let A denote the event X^2 > Y. Determine numerical values for the quantities E[XY] and E[XY|A].

Explanation / Answer

PMF of Y f(y) = 2/7 if y=0

= 2/7 if y=1

= 3/7 if y=2

= 0 o.w

PMF of X f(x) = 1/7 if x=0

= 2/7 if x=2

=1/7 if x=3

=2/7 if x=4

=1/7 if x=5

E(X) = 2*2/7 + 3*1/7 + 4*2/7 +5*1/7 =16/7 = 2.28

For x=2 E(y/x) is maximum

For y=2 V(x/y) is maximum

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