Question: Suppose 1 toss coins. Some of the them land heads and some land tails, Those that land tails 1 toss again. Let X be the number of heads showing after the first tossing, Y the total number showing after the second tossing, including the X heads appearing on the first tossing so X and Y are random variables such that 0 X Y 3no matter how the coins land. Write out distribution tables and sketch histograms for each of the following distributions:

a) The distribution of X:

b) The conditional distribution of Y given X = x for x = 0, 1, 2, 3:

c) The joint distribution of X and Y (no histogram in this case):

d) The distribution of Y:

e) The conditional distribution of X given Y = y for y = 0, 1, 2, 3.

f) What is the best guess of the value of X given Y = y for y = 0, 1, 2, 3? That is, for each y, choose x depending on y to maximize P(X = x|Y = y).

g) Suppose the random experiment generating X and Y is repeated independently over and over again. Each time you observe the value of Y, and then guess the value of X using the rule found in f). over the long run, what proportion of times will you guess correctly?