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.