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  3. suppose that \(y\) follows a binomial distribution

Suppose that \(Y\) follows a binomial distribution with parameters \(n\) and \(p\), so that the...

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Accounting

Suppose that \(Y\) follows a binomial distribution with parameters \(n\) and \(p\), so that the p.m.f. of given \(p\) is:

\[f(Y|p)= {n \choose y} p^y (1-p)^{n-y}.\]

Suppose that the prior p.d.f. of the parameter \(p\) is Beta\((a,b)\).

1.(10 points) Find the posterior distribution.

2.(10 points) Estimate the parameter \(p\), \(\hat p_{B}\) using the squared error loss function.

3.(10 points) Compare the Bayesian estimator from part b, \(\hat p_{B}\) to the MLE estimator from last homework, \(\hat p_{MLE}\).

4. (10 points) Now suppose that you run 20 trials, and got 8 successes. Use R and two different Beta priors to calculate the value of \(\hat p_{B}\). Do your estimate change a lot based on different priors?

5.(10 points) Calculate a Bayesian credible interval for \(p\) when \(\alpha=0.05\).

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