a) Suppose that $X$ is bounded. Justify that $X$ satisfies $(C_{\tau})$ for all $\tau$ in $\mathbb{R}^{+*}$. b) Suppose that $X$ follows the geometric distribution with parameter $p \in ]0,1[$ $$\forall k \in \mathbb{N}^{*}, \quad P(X = k) = p(1-p)^{k-1}$$ What are the real numbers $t$ such that $E\left(\mathrm{e}^{tX}\right) < +\infty$? For these $t$, give a simple expression for $E\left(\mathrm{e}^{tX}\right)$. c) Suppose that $X$ follows the Poisson distribution with parameter $\lambda$: $$\forall k \in \mathbb{N}, \quad P(X = k) = \mathrm{e}^{-\lambda} \frac{\lambda^{k}}{k!} \quad \text{where } \lambda \in \mathbb{R}^{+*}$$ What are the real numbers $t$ such that $E\left(\mathrm{e}^{tX}\right) < +\infty$? For these $t$, give a simple expression for $E\left(\mathrm{e}^{tX}\right)$.
a) Suppose that $X$ is bounded. Justify that $X$ satisfies $(C_{\tau})$ for all $\tau$ in $\mathbb{R}^{+*}$.
b) Suppose that $X$ follows the geometric distribution with parameter $p \in ]0,1[$
$$\forall k \in \mathbb{N}^{*}, \quad P(X = k) = p(1-p)^{k-1}$$
What are the real numbers $t$ such that $E\left(\mathrm{e}^{tX}\right) < +\infty$? For these $t$, give a simple expression for $E\left(\mathrm{e}^{tX}\right)$.
c) Suppose that $X$ follows the Poisson distribution with parameter $\lambda$:
$$\forall k \in \mathbb{N}, \quad P(X = k) = \mathrm{e}^{-\lambda} \frac{\lambda^{k}}{k!} \quad \text{where } \lambda \in \mathbb{R}^{+*}$$
What are the real numbers $t$ such that $E\left(\mathrm{e}^{tX}\right) < +\infty$? For these $t$, give a simple expression for $E\left(\mathrm{e}^{tX}\right)$.