91Ó°ÊÓ

An unknown parameter\(\theta \)needs to be estimated using a random sample\({X_1}...{X_n}\)of size n from the distribution for which\(\theta \)is the parameter.

Let\(\xi \left( \theta \right)\)be the prior distribution\(\theta \) and\(f\left( {x|\theta } \right)\)be the joint pdf of\({X_1}...{X_n}\).

The conditional pdf of \(\theta \) given \({X_1} = {x_1},...{X_n} = {x_n}\) is called the posterior distribution of \(\theta \) and is denoted by \(\xi \left( {\theta |{X_1}...{X_n}} \right)\). Then \(\xi \left( {\theta |{X_1}...{X_n}} \right) \propto {f_n}\left( {X|\theta } \right)\xi \left( \theta \right)\)

Let\(\theta \)be the proportion of defective items in a large manufactured lot.

A random sample of\(n = 8\)times yielded exactly 3 defectives.

Let\(\xi \left( \theta \right)\)be the prior distribution\(\theta \)

\(\xi \left( \theta \right) = \left\{ \begin{aligned}{l}2\left( {1 - \theta } \right)\;\;\;\;for\;0 < \theta < 1\\0\;\;\;\;\;\;\;\;\;\;\;\;\;otherwise\end{aligned} \right.\)

Let\({X_i} = 0\)if the inspected item is defective and\({X_i} = 1\)is the item is not defective.

Comparing the situation with Bernoulli distribution with parameter\(\theta \). Further, as each of the inspected items is independent of the other.

Hence joint pdf of \({X_1}...{X_n}\) is given by:

\(\begin{aligned}{f_n}\left( {X|\theta } \right) = \prod\limits_{i = 1}^8 {{\theta ^{\left( {{x_i}} \right)}}{{\left( {1 - \theta } \right)}^{\left( {1 - {x_i}} \right)}}} \\ = {\theta ^3}{\left( {1 - \theta } \right)^5}\end{aligned}\)

Consider the product of the joint pdf of\({X_1}...{X_n}\)and the prior pdf of\(\theta \)

\(\begin{aligned}{f_n}\left( {X|\theta } \right)\xi \left( \theta \right) = {\theta ^3}{\left( {1 - \theta } \right)^5}2\left( {1 - \theta } \right)\\ = 2{\theta ^{4 - 1}}{\left( {1 - \theta } \right)^{7 - 1}}\end{aligned}\)

We know that:

\(\begin{aligned}\xi \left( {\theta |{X_1}...{X_n}} \right) \propto {f_n}\left( {X|\theta } \right)\xi \left( \theta \right)\\ \propto 2{\theta ^{4 - 1}}{\left( {1 - \theta } \right)^{7 - 1}}\\{\theta ^{4 - 1}}{\left( {1 - \theta } \right)^{7 - 1}}\end{aligned}\)

Comparing the above expression with beta pdf with parameters\(\alpha = 4\)\(\beta = 7\), it’s clear that the conditional distribution of\(\theta \)is proportional to the beta distribution. That

is \(\frac{{\left| \!{\overline {\, {\left( {\alpha + \beta } \right)} \,}} \right. }}{{\left| \!{\overline {\, \alpha \,}} \right. \left| \! {\overline {\, \beta \,}} \right. }}\)

If multiplied with the constant term, the posterior pdf \(\theta \)will be beta with\(\alpha = 4\)and\(\beta = 7\). Thus, the posterior pdf of\(\theta \)is given by

\(\xi \left( {\theta |{X_1}...{X_n}} \right) = \frac{{\left| \!{\overline {\, {\left( {4 + 7} \right)} \,}} \right. }}{{\left| \!{\overline {\, 4 \,}} \right. \left| \!{\overline {\, 7 \,}} \right. }}{\theta ^{4 - 1}}{\left( {1 - \theta } \right)^{7 - 1}}\)