91Ó°ÊÓ

The proportion \(\theta \) of defective items in a large shipment is unknown and the prior distribution of \(\theta \) is the beta distribution with parameters 2 and 200

Since the observed number of defective items is 3 and the observed number of non-defective items is 97.

By the theorem, that P has the beta distribution with parameters\(\alpha \,\,and\,\,\beta \), and the conditional distribution of X given\(P = p\)is the binomial distribution with parameters n and p. Then the conditional distribution of P given\(X = x\)is the beta distribution with parameters\(\alpha + x\,\,and\,\,\beta + n - x\)

The posterior distribution of\(\theta \)is a beta distribution with parameters

\(\begin{array}{c}\alpha + x = 2 + 3\\ = 5\end{array}\)

And,

\(\begin{array}{c}\beta + n - x = 200 + 100 - 3\\ = 200 + 97\\ = 297\end{array}\)

Therefore, the posterior distribution of\(\theta \)is a beta distribution with parameters 5,297.