2. Suppose that observations y1, y2, . . . , yn are conditionally independent with yi Poisson (xi), where the values xi are known positive values of an explanatory variable, x, and is the unknown parameter of interest. Assume a gamma prior for , i.e., Gamma(, ). (a) Write down the likelihood function for . 1 (b) Derive the posterior distribution for . Show that the posterior distribution is a gamma distribution. Identity the shape and the rate parameters of this gamma distribution