For each of the following real random variables, determine the strictly positive reals $\alpha$ such that the random variable admits an exponential moment of order $\alpha$ and calculate $\mathbb{E}\left(\mathrm{e}^{\alpha X}\right)$ in this case. a) $X$ a random variable following a Poisson distribution with parameter $\lambda$, where $\lambda$ is a strictly positive real. b) $Y$ a random variable following a geometric distribution with parameter $p$, where $p$ is a real strictly between 0 and 1. c) $Z$ a random variable following a binomial distribution with parameters $n$ and $p$, where $n$ is a strictly positive integer and $p$ is a real strictly between 0 and 1.
For each of the following real random variables, determine the strictly positive reals $\alpha$ such that the random variable admits an exponential moment of order $\alpha$ and calculate $\mathbb{E}\left(\mathrm{e}^{\alpha X}\right)$ in this case.
a) $X$ a random variable following a Poisson distribution with parameter $\lambda$, where $\lambda$ is a strictly positive real.
b) $Y$ a random variable following a geometric distribution with parameter $p$, where $p$ is a real strictly between 0 and 1.
c) $Z$ a random variable following a binomial distribution with parameters $n$ and $p$, where $n$ is a strictly positive integer and $p$ is a real strictly between 0 and 1.