91Ó°ÊÓ

The number of defects in a 1200-foot roll of magnetic recording tape has a Poisson distribution with a mean of\(\theta \)is unknown.

The prior distribution of \(\theta \) is the gamma distribution with parameters \(\alpha = 3\,\,and\,\,\beta = 1\)

Let,\(\alpha \,\,and\,\,\beta \)are positive numbers.

A random variable X has the gamma distribution with parameters\(\alpha \,\,and\,\,\beta \)

The probability density function of X is,

\(f\left( {x\left| {\alpha ,\beta } \right.} \right) = \left\{ {\begin{aligned}{\frac{{{\beta ^\alpha }}}{{\Gamma \left( \alpha \right)}}{x^{\alpha - 1}}{e^{ - \beta x}}}&{for\,\,x > 0,}\\0&{otherwise.}\end{aligned}} \right.\)

Sampling from a Poisson distribution,

Suppose\({X_1},...,{X_n}\)from a random sample from a Poisson distribution with mean\(\theta > 0,\,\,and\,\theta \)unknown. Suppose also that the prior distribution of\(\theta \)is the gamma distribution with parameters\(\alpha > 0\,\,and\,\,\beta > 0\). Then the posterior distribution of\(\theta \)given that\({X_i} = {x_i}\,\,\left( {i = 1,...,n} \right)\)is the gamma distribution with parameters\(\alpha + \sum\limits_{i = 1}^n {{x_i}} \,\,and\,\,\beta + n\).

Suppose that the mean defects in a 1200-foot roll of magnetic recording tape have a Poisson distribution for which the value of\(\theta \)is unknown and that the prior distribution of\(\theta \)is the gamma distribution with parameters\(\alpha = 3\,\,and\,\,\beta = 1\).

When the five rolls of this tape are selected at random and inspected, the numbers of defects found on the rolls are 2,2,6,0, and 3 (n=5)

The posterior distribution of\(\theta \)is,

Let X denote the number of defects on the rolls.

The posterior distribution of\(\theta \)with parameters is,

\(\begin{aligned}\left( {\alpha + \sum\limits_{i = 1}^5 {{x_i}} } \right) &= 3 + \left( {2 + 2 + 6 + 0 + 3} \right)\\ &= 3 + 13\\ &= 16\end{aligned}\)

And

\(\begin{aligned}{c}\left( {\beta + n} \right) = 1 + 5\\ = 6\end{aligned}\)

Therefore, the posterior distribution of \(\theta \) is the gamma distribution with parameters 16 and 6.