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Chapter 5: Q11E (page 302)

Prove that the p.f. of the negative binomial distribution can be written in the following alternative form:

\[{\bf{f}}\left( {{\bf{x|r,p}}} \right){\bf{ = }}\left\{ \begin{array}{l}\left( \begin{array}{l}{\bf{ - r}}\\{\bf{x}}\end{array} \right){{\bf{p}}^{\bf{r}}}{\left( {{\bf{ - }}\left[ {{\bf{1 - p}}} \right]} \right)^{\bf{x}}}\;\;{\bf{for}}\,{\bf{x = 0,1,2}}...\\{\bf{0}}\;\;\;{\bf{otherwise}}{\bf{.}}\end{array} \right.\]Hint: Use Exercise 10 in Sec. 5.3.

Short Answer

Expert verified

Negative binomial distribution can also be written in the form:

\(f\left( {x|r,p} \right) = \left\{ \begin{array}{l}\left( \begin{array}{l} - r\\x\end{array} \right){p^r}{\left( { - \left[ {1 - p} \right]} \right)^x}\;\;for\,x = 0,1,2...\\0\;\;\;{\rm{otherwise}}{\rm{.}}\end{array} \right.\)

Step by step solution

01

Given information.

X=0,1,2 follows negative binomial distribution.

02

Computing pf of negative binomial distribution.

According to exercise 10.

\(\left( \begin{array}{l} - r\\x\end{array} \right) = {\left( { - 1} \right)^x}\left( \begin{array}{l}r + x - 1\\x\end{array} \right)\)

This makes

\(\left( \begin{array}{l} - r\\x\end{array} \right){p^r}{\left( { - \left[ {1 - p} \right]} \right)^x} = \left( \begin{array}{l}r + x - 1\\x\end{array} \right){p^r}{\left( {1 - p} \right)^x}\)

Which is the proper form of the negative binomial pf for x=0,1,2...

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Let \({{\bf{X}}_{{\bf{1,}}}}{\bf{ }}{\bf{. }}{\bf{. }}{\bf{. , }}{{\bf{X}}_{\bf{n}}}\)be i.i.d. random variables having thenormal distribution with mean \({\bf{\mu }}\) and variance\({{\bf{\sigma }}^{\bf{2}}}\). Define\(\overline {{{\bf{X}}_{\bf{n}}}} {\bf{ = }}\frac{{\bf{1}}}{{\bf{n}}}\sum\limits_{{\bf{i = 1}}}^{\bf{n}} {{{\bf{X}}_{\bf{i}}}} \) , the sample mean. In this problem, weshall find the conditional distribution of each \({{\bf{X}}_{\bf{i}}}\)given\(\overline {{{\bf{X}}_{\bf{n}}}} \).

a.Show that \({{\bf{X}}_{\bf{i}}}\)and\(\overline {{{\bf{X}}_{\bf{n}}}} \) have the bivariate normal distributionwith both means \({\bf{\mu }}\), variances\({{\bf{\sigma }}^{\bf{2}}}{\rm{ }}{\bf{and}}\,\,\frac{{{{\bf{\sigma }}^{\bf{2}}}}}{{\bf{n}}}\),and correlation\(\frac{{\bf{1}}}{{\sqrt {\bf{n}} }}\).

Hint: Let\({\bf{Y = }}\sum\limits_{{\bf{j}} \ne {\bf{i}}} {{{\bf{X}}_{\bf{j}}}} \).

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