Solution Found!
We saw in Exercise 5.159 that the negative binomial random
Chapter 6, Problem 58E(choose chapter or problem)
We saw in Exercise 5.159 that the negative binomial random variable Y can be written as \(Y=\sum_{i=1}^{r} W_i\), where \(W_{1}, W_{2}, \ldots, W_{r}\) are independent geometric random variables with parameter p.
a. Use this fact to derive the moment-generating function for Y.
b. Use the moment-generating function to show that \(E(Y)=r / p\) and \(V(Y)=r(1-p) / p^{2}\)
c. Find the conditional probability function for \(W_1\), given that \(Y=W_{1}+W_{2}+\cdots+W_{r}=m\)
Questions & Answers
QUESTION:
We saw in Exercise 5.159 that the negative binomial random variable Y can be written as \(Y=\sum_{i=1}^{r} W_i\), where \(W_{1}, W_{2}, \ldots, W_{r}\) are independent geometric random variables with parameter p.
a. Use this fact to derive the moment-generating function for Y.
b. Use the moment-generating function to show that \(E(Y)=r / p\) and \(V(Y)=r(1-p) / p^{2}\)
c. Find the conditional probability function for \(W_1\), given that \(Y=W_{1}+W_{2}+\cdots+W_{r}=m\)
ANSWER:Step 1 of 4:
It is given that \(W_{1}, W_{2}, W_{3}\),... are independent Geometric random variables with parameter \(p\).
Also it is given that \(Y\) is a negative binomial random variable and \(Y\) is expressed as
\(Y=\sum_{i=1}^rW_i\)
Using this we have to obtain the moment generating function of \(Y\) and we have to show that
\(\mathrm{E}(\mathrm{Y})=\frac{r}{p}, \mathrm{~V}(\mathrm{Y})=\frac{r(1-p)}{p^{2}}\)
Also, we have to derive the conditional probability function.