Consider the Poisson distribution with parameter μ, where μ > 0. Its probability mass function is given by:
pμ(y) = e^-μμ^y/y!
for y = 0, 1, 2, ...
(a) Show that the Poisson distribution belongs to the exponential family of distribution with natural parameter η = log(μ), where log(x) = ln(x), and show that its cumulant function is c(η) = e^η.
(b) What is the carrier measure of the distribution?
(c) Use the cumulant function to show that the mean and the variance for this distribution are E[Y] = μ, and V[Y] = μ, respectively.
(d) Suppose that we have a random sample of size n = 55 from a Poisson distribution with mean μ. The sample mean of the 55 observations is ȓ = 15. Use the score test to test
H0 : μ = 20 against Ha : μ ≠ 20.
Give the observed value of the score test statistic, the p-value, and the conclusion at α = 5%.