1. Suppose X has a binomial distribution with parameters n and p. Then its moment-generating function...

1. Suppose X has a binomial distribution with parameters n and p. Then its moment-generating function is M(t) = (1 − p + pe^t ) n .

(a) Use the m.g.f. to show that E(X) = np and Var(X) = np(1 − p).

(b) Prove that the formula for the m.g.f. given above is correct. Hint: the binomial theorem says that Xn x=0 n x a^x b^(n−x) = (a + b)^n .

2. Suppose X has a Poisson distribution with parameter λ. Then its moment-generating function is M(t) = e^(λ(e t−1)) .

(a) Use the m.g.f. to show that E(X) = λ and Var(X) = λ.

(b) Prove that the formula for the m.g.f. given above is correct. Hint: for any real number a, a^x / x! = e a .

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orchestra answered 1 year ago

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