8. For a large positive integer n, let X be a Poisson random variable with parameter n;

let Y be a gamma random variable with parameters n and λ, and let W be a binomial random variable with parameters n and p. Show that X is approximately N(n, n), Y is approximately N(n/λ, n/λ2), andW is approximately N

????

np, np(1 − p)

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Hint: Note that X is the sum of n independent Poisson random variables, each with parameter 1; Y is the sum of n independent exponential random variables, each with parameter λ; andW is the sum of n independent Bernoulli random variables, each with parameter p.