The negative binomial random variable Y was defined in

QUESTION:

The negative binomial random variable \(Y\) was defined in Section  as the number of the trial on which the \(r\)th success occurs, in a sequence of independent trials with constant probability \(p\) of success on each trial. Let \(X_i\) denote a random variable defined as the number of the trial on which the \(i\)th success occurs, for \(i=1,2,....r\). Now define

                            \(W_{i}=X_{i}-X_{i-1}, i=1,2, \ldots r\)

where \(X_0\) is defined to be zero. Then we can write \(Y=\sum_{i=1}^{r} W_{i}\). Notice that the random variables \(W_{1}, W_{2}, \ldots, W_{r}\), have identical geometric distributions and are mutually independent. Use Theorem  to show that \(E(Y)=r/p\) and \(V(Y)=r(1-p)/p^2\).

Equation Transcription:

Text Transcription:

Y

r

 p

X_i

i

i=1,2,....r

W_i=X_i-X_i-1, i=1,2,....r

X_0

Y=sum_i=1^r W_i

W_,W_2,...,W_r

E(Y)=r/p

V(Y)=r(1-p)/p^2