Negative binomial distribution - assume Bernoulli trials (e.x. keep tossing a co

Question

Negative binomial distribution - assume Bernoulli trials (e.x. keep tossing a coin as many times as we want) that is, (1) there are two possible outcomes, (2) the trials are independent, and (3) p is the probability of success for each trial, the probability of success, remains the same from trial to trial. Let X denote the number of trials until the r success. Then we say X follows a negative binomial distribution with parameters n and r. (a) write down the probability distribution of X. (b) Compare negative binomial distribution with geometric distribution. Find E(X) and V ar(X) based on the properties of geometric distribution.

Explanation / Answer

a)

X!/(X-r)!*p^r*(1-p)^X-r

b)

Geometric distribution

q=1-p

p(x)=pq^x

Mean (x)=q/p, Var(x)=q/p^2

Negative binomial distribution

p(x)={r+x-1

x}*p^r*q^x

Mean(x)=rq/p, Var(x)=rq/p^2

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