Question

Let N be the number of trials until k consecutive successes have occurred when each trial is independently a success with probability p.

(a) What is P(N = k)?
(b) Argue that the expected value of N is k/p.
(c) Argue that the variance of N is k(1-p)/p²."

Answer 1

(a) P(N = k): The probability that it takes exactly k trials to achieve k consecutive successes can be calculated using the formula P(N = k) = p^k * (1-p)^(k-1). (b) Expected value of N: The expected value or mean of N can be calculated using the formula E(N) = k/p. (c) Variance of N: The variance of N can be calculated using the formula Var(N) = k(1-p)/p^2.

Step-by-step explanation:

(a) P(N = k):

The probability that it takes exactly k trials to achieve k consecutive successes can be calculated using the formula P(N = k) = p^k * (1-p)^(k-1). This is because in order to have k consecutive successes on the kth trial, the first k-1 trials must each be a success followed by a failure on the kth trial.

(b) Expected value of N:

The expected value or mean of N can be calculated using the formula E(N) = k/p. This formula represents the average number of trials needed to achieve k consecutive successes.

(c) Variance of N:

The variance of N can be calculated using the formula Var(N) = k(1-p)/p^2. This formula represents the spread or variability in the number of trials needed to achieve k consecutive successes.