Two twin brothers, Hank and Crank are kicking a ball around in a park. Tim challenges James to a contest on who can kick the ball the furthest and so they agree to terms: two rounds, 10 kicks in each round.

Let:

X = the number of kicks in round 1 for which Crank'skicks travel farther than Hank 's and,
Y = the number of kicks in round 2 for which Crank'skicks travel farther than Hank 's.

We assume that X|p Binomial (10,p) and Y|p Binomial(10,p) are independent.

a) What is E(X + Y|p)? what is E((X + Y)/20|p)?

b) What is the distribution of X+ Y and why?

c) Suppose that the result of round 1 is X = 4 and the result of round 2 is Y = 2. Crank is understandably annoyed with this result, but perhaps not quite ready to concede defeat. Crank 'sprior density for possible p is Beta(10,10), i.e., p ~Beta(10,10). The Beta(a,b) density and expectation generally are:

find the posterior density for p|X+Y = 6, i.e. f_pX+Y=6 (p||X+Y = 6). what is the posterior mean E(p|X+Y = 6) and how does this compare to the data only estimate of p, 6/20 = 3/10?

d) Alternatively, suppose that p = 0.3. let W be the distance of Tim 's kicks on a standard scale between (0,1); let Z be the distance of James's kicks, also on (0,1). W and Z are independent with the following marginal densities:

f_W (w) = 3w², 0

f_Z (z) = tz^{^(t-1)}

what value of t makes P(Z>W) = p = 0.3?