A Poisson random variable may be used as approximation for a binomial random variable with parameter (n, p) when n is large and p is small so that np is of moderate size. To see this, let X ~ Bin (n, p) such that, lambda:= np is a positive constant. In other words, p is a function of n given by p = p_n = lambda/n. Show that for every non-negative integer k, P(X = k)= (n k) p^k_n (1 - p_n)^n - k rightarrow e^-lambda lambda^k/k! as n rightarrow infinity. (use that fact that (1 - lambda/n)^n rightarrow e^-lambda as n rightarrow infinity)