Question

2. Estimate the probability 0 of teen recidivism based on a study in which there were n- 43 individuals released from incarceration and y 15 re-offenders within 36 months. (a) Using a Beta(2,8) prior for θ, plot p(9), p(y|0) and p( ly) as functions of θ. Find the posterior mean, mode, and standard deviation of. Find a 95% quantile-based confidence interval. (b) Repeat part (a), but using a Beta(8,2) prior for θ (c) Consider the following prior distribution: 1 г(10) pe) rere) [39(1-0)" +gr(1-0)] , which is a 75-25% of a Beta(2,8) and a Beta(8,2) prior distribution. Plot this prior distribution and compare it to the priors in (a) and (b). Describe what sort of prior opinion this may represent. (d) For the prior in part (c) i. Write out mathematically p(0) x p(y9) and simplify as much as possible. ii. The posterior distribution is a mixture of two distributions you know. Identify these distributions. iii. Calculate and plot p(0) x P(y|θ) as a function of θ. Also find (approximately) the posterior mode, and discuss its relation to the modes in (a) and (b) (e) Find a general formula for the weights of the mixture distribution in (d)ii, and provide an interpretation for their values.

Answer 1