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 p(0, yoly) x 0 1-0 1-0 0 +- n! and thus, the sampling model is a multinomial distribution: n! p(yle) = y!y2!Y3!y4! 4' 4 4 where n = y +92 +93 +34. Assume the prior distribution for 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 (yo, y-yo) with probabilities (1). Here yo can be viewed as another parameter (or a latent variable). Thus, yo! (y1 - y0)!y2!y3!Y! 1-09) 1 ( + 9) " (9)" ( = )" (1) ". 4 17/0 yi-90 () () ())" 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 0 (plot the density). 3. Find the estimate and 95% credible interval of 0. Hint: 0 4 + 4 + : 1 -