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Let \(X\) be a random variable taking values \(n = 0,1,2,\ldots\) with probabilities \(p_n\). The function

\[ G(s) = \mathrm{E}[s^X] = p_0 + p_1 s + p_2 s^2 + \ldots, \quad |s| \leq 1 \]

is called the generating function of the random variable \(X\). Derive the following formulas:

(a) If \(X\) is a Poisson random variable, i.e., \(p_n = e^{-\lambda} \lambda^n / n!\) where \(\lambda > 0\) and \(n \in \mathbb{Z}_+\), then

\[ G(s) = e^{-\lambda(1-s)}, \quad |s| \leq 1 \]

(b) If \(X\) has a geometric distribution, i.e., \(p_n = p(1-p)^n\) where \(0 < p < 1\) and \(n \in \mathbb{Z}_+\), then

\[ G(s) = \frac{p}{1-s(1-p)}, \quad |s| \leq 1 \]

(c) If \(X_1, \ldots, X_n\) are i.i.d. random variables with

\[ \mathrm{P}(X_1 = 1) = p \quad \mathrm{P}(X_1 = 0) = 1 - p \]

and \(X = X_1 + \ldots + X_n\), then

\[ G(s) = (ps + 1 - p)^n = \sum_{k=0}^{n} \binom{n}{k} p^k (1-p)^{n-k} s^k \]

and, therefore, \(\mathrm{P}(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, \quad 0 \le k \le n\).

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