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Chapter 7: Q3E (page 405)

Suppose that the proportion 胃 of defective items in a large shipment is unknown and that the prior distribution of 胃 is the beta distribution with parameters 2 and 200. If 100 items are selected at random from the shipment and if three of these items are found to be defective, what is the posterior distribution of 胃?

Short Answer

Expert verified

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

Step by step solution

01

Given information

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

02

Finding the posterior distribution of \({\bf{\theta }}\) 

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.

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Most popular questions from this chapter

Question:It is not known what proportion p of the purchases of a certain brand of breakfast cereal are made by women and what proportion are made by men. In a random sample of 70 purchases of this cereal, it was found that 58 were made by women and 12 were made by men. Find the M.L.E. of p.

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