Let the random variable X denote the number of trials in ecess of r that are ed to achieve the r-th success in a series of independent trials, where p is the pility of success at any given trial. Show that

Given that,

The random variable X as the number of trails in excess of r that are required to achieve the r-th success.

The probability of success for given trail is p and the probability of failure is 1-p.

Let us consider, the first success, r=1

P(X=0)=p

P(X=1)=p(1-p)

P(X=2)= p(1-p)² and so on….

Let us consider, the second success, r=2

P(X=0)=p²

P(X=1)=2p²(1-p)

P(X=2)=3p²(1-p)² and so on….

Let us consider, the r-th success,

P(X=0)=p^r

P(X=1)=rp^r(1-p)

P(X=2)=[(r+1)r/2]p²(1-p)² and so on….