91Ó°ÊÓ

The probability mass function is given as

\({\rm P}\left( {X = x} \right) = \left( \begin{align}x - 1\\r - 1\end{align} \right){p^r}{\left( {1 - p} \right)^{x - r}}\)

For\(x = r,r + 1,r + 2...\)

Let\({X_1},...{X_n}\)be the random sample from the specified distribution with parameter r and p

The likelihood function for \({X_1}...{X_n}\) is,

\(\begin{array}L\left( {{X_1}...{X_n}|p} \right) = f\left( {{X_1}...{X_n}} \right)\\ = f\left( {{X_1}} \right) \times f\left( {{X_2}} \right) \times ... \times f\left( {Xn} \right)\\ = \left[ {\left( \begin{array}{l}{x_1} - 1\\r - 1\end{array} \right){p^r}{{\left( {1 - p} \right)}^{{x_1} - r}}} \right] \times ... \times \left[ {\left( \begin{array}{x_n} - 1\\r - 1\end{array} \right){p^r}{{\left( {1 - p} \right)}^{{x_n} - r}}} \right]\end{array}\)

The resultant function is in the form of

\(g\left( {T,p} \right) \times h\left( {{x_1},...{x_n}} \right)\)

Here

\(g\left( {T,p} \right) = \left( {{p^{nr}}{{\left( {1 - p} \right)}^{\sum\limits_{i = 1}^n {{x_i} - np} }}} \right)\) and

\(h\left( {{x_1},...{x_n}} \right) = \prod\limits_{i = 1}^n {\left( \begin{align}{l}{x_1} - 1\\r - 1\end{align} \right)} \)

Hence,

\(T = \sum\limits_{i = 1}^n {{x_i}} \) is sufficient statistic for p.