A coin is flipped until heads occur twice. Define two random variables X and Y to be the trial numbers at which the first and the second heads are observed. Assume that at any trial, the probability...

A coin is flipped until heads occur twice. Define two random variables X and Y to be the trial numbers at which the first and the second heads are observed. Assume that at any trial, the probability


A coin is flipped until heads occur twice. Define two random variables X and Y to be the trial numbers at which the first and the second heads are observed. Assume that at any trial, the probability that a head occurs is p is between (0, 1). Where probability of tail is q=1-p (1). Show that the joint probability is given by P(X = m, Y = n) = ( p2qn-2 , where m = 1, 2, . . . ; and n = 2, 3, . . . , 0 , otherwise.) (2) Calculate the marginal probability mass function for X and Y. (3) Calculate the conditional probability that X = m, given Y = n (i.e., P(X =m | Y=n)