Question: Let (X) be a (operatorname{Poisson}(lambda)) random variable. a. Prove that the moment generating function of (X) is (exp [lambda exp (t)-1]). b. Prove that the
Let \(X\) be a \(\operatorname{Poisson}(\lambda)\) random variable.
a. Prove that the moment generating function of \(X\) is \(\exp [\lambda \exp (t)-1]\).
b. Prove that the characteristic function of \(X\) is \(\exp [\lambda \exp (i t)-1]\).
c. Using the moment generating function, derive the first three moments of \(X\). Repeat the process using the characteristic function.
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