Let y_ij be observation j of a count variable for group i, i = 1,...,I, j = 1,..., n_i. Suppose that {Y_ij} ae independent Poisson with E(Y_ij) = µ_i.

a. Show that the ML estimate of µ_i is µ̂_i = y̅_i = ∑_j y_ij/n_i,

b. Simplify the expression for the deviance for this model. [For testing this model, it follows from Fisher that the deviance and the Pearson statistic ∑_i ∑_j (y_ij – y̅_i)²/y̅_i have approximate chi-squared distributions with df = ∑_i(n_i – 1). For a single group, Cochran (1954) referred to ∑_j(y_1j – y̅₁)²/y̅₁ as the variance test for the fit of a Poisson distribution, since it compares the sample variance to the estimated Poisson distribution, since it compares the sample variance to the estimated Poisson variance y̅₁.]