Background When working with genomic data, the Bernoulli distribution can be used to model the number of sequencing reads mapped to a particular gene or transcript during a sequencing process. Suppose that $1, 12. ..., I, ~ Bernoulli (@) and that o has a Beta(a, 8) prior, such that the prior distri- bution of 0 is given by 1 p(#) = B(a, B) To)r(B) where B(a, B) = Recall also that the pmf of X|0 ~ Bernoulli(0) is given by [(a + B) 05 (1 - 0)1-1. Questions Using this information, complete the following questions. Make sure to show all your workings. (a) Find the expression for the maximum likelihood estimate (MLE) of d. (b) Determine (up to a constant term) the probability density function of the posterior distribution of e. Hint: You may use the fact that a! = D(a -1) for some positive integer a. (c) Show that the posterior estimate of o is a weighted average between the MLE and the prior estimate of e under the Beta prior. Hint: You may use the fact that the prior estimate of d is