Let X denote the random variable representing the number of defective items in the random sample of 20 items.
Now, we are given that the prior distribution of is the beta distribution with parameters 1 and 10
=> ~ Beta(α=1, β=10)
Moreover, since is the proportion of defective items in the random sample of 20 items, we can conclude that:
X| ~ Binomial(n=20, )
(a)
Now, the expected value and variance of prior distribution is given by:
(b)
To find the posterior distribution, we use the following relation:
Posterior Prior*Likelihood ................(1)
Now, we are given the following details about the prior:
~ Beta(α=1, β=10)
Also, we are given that X|
~ Binomial(n=20,
) and that we observed X = 1 (since in a random sample of 20 items
we found 1 defective item). Thus:
Now, substituting the value of Prior and Likelihood in equation (1), we get:
(c)
The Bayes estimator for under the quadratic loss function is just the mean of the Posterior disribution, thus we get:
(d)
The likelihood function is given by:
To check that the last expression for is indeed a point of maxima for the likelihood, we again differentiate expression (2) w.r.t. :
Thus, the MLE of is:
From part (c) and the last expression we see that the Bayes estimator under quadratic loss and the M.L.E. for are not the same. [ANSWER]
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