Derive MLE and Bayesian posterior for Bernoulli

Q: Derive MLE and Bayesian posterior for Bernoulli

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Question

Bernoulli/Binomial Inference Task

You observe n independent Bernoulli trials with unknown success probability p, and you record k successes (so K ~ Binomial(n, p)).

Tasks

(a) Derive the maximum likelihood estimator (MLE) of p and its asymptotic variance.

(b) Assume a Beta(alpha, beta) prior on p. Derive the posterior distribution of p and the posterior predictive probability that the next trial is a success.

(c) Compute a 95% confidence interval (CI) for p using the normal approximation, and a 95% credible interval from the posterior in (b).

(d) Explain when each interval (Wald CI vs. Bayesian credible interval) is reliable and how sample size affects the inference.

Derive MLE and Bayesian posterior for Bernoulli

Bernoulli/Binomial Inference Task

Tasks

Solution (Locked)

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