3. Genetic study. A genetic study has divided n = 197 animals into four categories: y = (125, 18, 20, 34). A genetic model for the population cell probabilities is given by 1- NI - 1 ) and thus, the sampling model is a multinomial distribution: n! p(yle) = yilyzyslya! (3+ 9 ) " ( 4 9 ) " ( 49 ) " ( 4 ) " 2 where n = 31 + 92 + 93 + y. Assume the prior distribution for e to be Uniform(0, 1). To find the posterior distribution of 0, a Gibbs sampling algorithm can be implemented by splitting the first category into two (30, 31 - 30) with probabilities (-, =). Here yo can be viewed as another parameter (or a latent variable). Thus, n! p(e, yoly) o yo!(31 - yo)!yz!yay! 1. Derive the full conditional distributions of 0 and yo- 2. Implement Gibbs sampling in R. Matlab, Python, or Winbugs and obtain the posterior distribution of 8 (plot the density). 3. Find the estimate and 95% credible interval of 9. Hint: + + +12 191-1