Expected number of trials

Question

Consider independent trials, each of which is a success with probability p and derive the expected number of trials needed to obtain k consecutive successes by

(a)conditioning on the time of the first failure

(b)conditioning on the time that it takes to obtain k-1 consecutive successes

ANS.

I am thinking the answer to this question. but still I am unable.

Answer 1

Hint:-Here trials follows Negative Binomial Distribution.(type-2 & type-1 respectively)

So then, (a) $\frac{P(success)}{P(failure)}$

(b) $\frac{1}{P(failure)}$

Answer 2

Let us solve part (b) of question first. Here X is geometric with parameter p if

P{X=k}=$p^{k-1}q,$, k=1,2,... where q=1-p Hence, its moment generating function is $\phi(t)=E[e^{tx}]$

$=\displaystyle\sum_{k=1}^{\infty}e^{tk}p^{k-1}q$

$=qe^t\displaystyle\sum_{k=1}^{\infty}(pe^t)^{k-1}$

$=\frac{qe^t}{1-pe^t}$

Differentiation gives

$\phi'(t)=\frac{(1-pe^t)qe^t+qe^tpe^t}{(1-pe^t)^2}$ $=\frac{qe^t}{1-pe^t}$ Evaluation at t=0 gives

E[X]=$\frac{q}{(1-p)^2}=\frac1q$

If we solve part(a) in this way, we get the answer $\frac{p}{q^2}$