Assume you throw a dart at a target and assume the throws are independent of each other. Each time you throw a dart, the probability that the dart hits the target is p. Assume you will throw the dart until the dart hits the target twice (e.g., you will stop throw the dart at the 5th throw, if the dart hit the target for the first time in the 3rd throw and hit the target for the 2nd time in the 5th throw). Denote X and Y as the numbers of times until the dart hits the target for the first and the second time.
(a) Show that the joint pmf of X and Y is pX,Y(i,j) = p^2(1-p)^(j-2), for 1<=i
(b) Find the marginal pmfs of X and Y.
(c) Are X and Y ind ify your answer.
(d) Find the conditional pmf of X given Y. What type of distribution is this conditional pmf?
(e) Find the conditional pmf of Y given X.