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
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My question is attached.
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)=pr
P(X=1)=rpr(1-p)
P(X=2)=[(r+1)r/2]p2(1-p)2 and so on….
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