91Ó°ÊÓ

Consider an experiment with only two results.

Let X be the number of trials required until first success is obtained.

Let P be the probability of getting desired result in a single trial.

Then X is said to follow geometric distribution and the probability mass function is \({\rm P}\left( {X = x} \right) = p{\left( {1 - p} \right)^x}\;;\;x \ge 1\)

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

Let T be the statistic.

If for every parameter \(\theta \) and for all possible values of the conditional joint distribution of \({X_1},...{X_n}\) given\(T = t\), is only depend on the values of t but not on the parameter \(\theta \) and this condition satisfy for all \(\theta \) then T can be a sufficient statistic for the parameter \(\theta \)

Let \({X_1},...{X_n}\) be the random sample from the specified distribution with unknown parameter \(\theta \in \Omega \).with the pdf \(f\left( {x|\theta } \right)\)

\(T = r\left( {{X_1},...{X_n}} \right)\) is sufficient statistic if and only if joint pdf \(f\left( {x|\theta } \right)\) can be factored as \({f_n}\left( {x|\theta } \right) = u\left( x \right)v\left{ {r\left( x \right),\theta } \right)\)

Let \(X = {X_1},...{X_n}\)are the random samples form the Bernoulli distribution

The pdf is \(f\left( {x,p} \right) = \left\{ \begin{align}p{\left( {1 - p} \right)^x}\;\;\;for\;x = 0,1\\0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;otherwise\end{align} \right.\)

T be the value of statistic when the observed values of \(X = {X_1},...{X_n}\)are \(x = {x_1},{x_2}...{x_n}\)

The joint pdf is given as:

\(\begin{align}{f_n}\left( {x,p} \right) &= p{\left( {1 - p} \right)^{{x_1}}} \times p{\left( {1 - p} \right)^{{x_2}}}...p{\left( {1 - p} \right)^{{x_n}}}\\{f_n}\left( {x,p} \right) &= {p^n}{\left( {1 - p} \right)^{\sum\limits_{i = 1}^n {{x_i}} }}\\{f_n}\left( {x,p} \right) &= {p^n}{\left( {1 - p} \right)^{r\left( x \right)}}\end{align}\)

Using factorization theorem assume that\(p > 0\)and from the joint pdf it has

\(u\left( x \right) = 1\)and

\(v\left{ {r\left( x \right),\theta } \right) = {p^n}{\left( {1 - p} \right)^{\sum\limits_{i = 1}^n {{x_i}} }}\)

Where\(r\left( x \right) = \sum\limits_{i = 1}^n {{x_i}} \).

Therefore, the statistic\(T = \sum\limits_{i = 1}^n {{x_i}} \)is sufficient statistic.