Test the convergence of Theorem 2.8. For α = 10, plot the PMF of Kn for (n, p) = (10, 1), (n, p) = (100, 0.1), and (n, p) = (1000, 0.01) and compare against the Poisson (α) PMF.
Theorem 2.8
Perform n Bernoulli trials. In each trial, let the probability of success be α/n, where α > 0 is a constant and n > α. Let the random variable Kn be the number of successes in the n trials. As n → ∞, PKn (k) converges to the PMF of a Poisson (α) random variable.