# Setting up likelihood function and finding MLE

Consider the following data from 50 Bernoulli Trials

Let theta be the probability of the observing a one for any single trial.

a) Find the likelihood of any sequence of s trials containing y ones.

b) If the prior on theta is uniform such that p(theta)=1, find an expression for the posterior distribution for theta after 10 trials. What is the maximum likelihood estimator of theta?

c) Find an expression of the posterior after another 30 trials. In what sense has the prior been updated?

See attached for data

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#### Solution Preview

a) Find the likelihood of any sequence of s trials containing y ones.

The likelihood based on the bernoulli distribution is p(y|θ)=θy(1-θ)s-y

b) If the prior on theta is uniform such that p(theta)=1, find an expression for the posterior ...

#### Solution Summary

The solution gives detailed steps on setting up likelihood function, finding MLE and then discussing properties. It involves finding the likelihood of any sequence of s trials, finding maximum likelihood estimator of theta, and finding the expression of a posterior.