Question: 24. A device fails to operate on a single trial with probability p. Let X be a random variable denoting the number of trials required

24. A device fails to operate on a single trial with probability p. Let X be a random variable denoting the number of trials required to obtain a total of k failures. If results of successive trials are independent, show that the probability function of X is given by p(x) =

x − 1 k − 1

pkqx−k; x = k, k + 1, . . .

Show that the mean and variance of X are given by E(X) = k p

, Var(X) = kq p2

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