We suppose here that the infimum $\eta_{a}$ of the function $\chi$ on $I \cap \mathbb{R}^{+}$ is attained at a point $\sigma$ interior to $I \cap \mathbb{R}^{+}$. Let $t$ be a real number interior to $I$ and such that $t > \sigma$, $b$ be a real number such that $b > \frac{\varphi_{X}^{\prime}(t)}{\varphi_{X}(t)}$. a) Calculate $\sum_{x \in X(\Omega)} \frac{\mathrm{e}^{tx}}{E\left(\mathrm{e}^{tX}\right)} P(X = x)$. We then admit (if necessary by modifying $(\Omega, \mathcal{A}, P)$) that there exists a random variable $X^{\prime}$ on $(\Omega, \mathcal{A})$ such that $X^{\prime}(\Omega) = X(\Omega)$ and whose probability distribution is given by $$\forall x \in X(\Omega), \quad P\left(X^{\prime} = x\right) = \frac{\mathrm{e}^{tx}}{E\left(\mathrm{e}^{tX}\right)} P(X = x)$$ and that there exists a sequence $\left(X_{n}^{\prime}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables defined on $(\Omega, \mathcal{A}, P)$ all following the same distribution as $X^{\prime}$. b) Show $$E\left(X^{\prime}\right) = \frac{\varphi_{X}^{\prime}(t)}{\varphi_{X}(t)}, \quad E\left(X^{\prime}\right) > a$$
We suppose here that the infimum $\eta_{a}$ of the function $\chi$ on $I \cap \mathbb{R}^{+}$ is attained at a point $\sigma$ interior to $I \cap \mathbb{R}^{+}$. Let $t$ be a real number interior to $I$ and such that $t > \sigma$, $b$ be a real number such that $b > \frac{\varphi_{X}^{\prime}(t)}{\varphi_{X}(t)}$.
a) Calculate $\sum_{x \in X(\Omega)} \frac{\mathrm{e}^{tx}}{E\left(\mathrm{e}^{tX}\right)} P(X = x)$.
We then admit (if necessary by modifying $(\Omega, \mathcal{A}, P)$) that there exists a random variable $X^{\prime}$ on $(\Omega, \mathcal{A})$ such that $X^{\prime}(\Omega) = X(\Omega)$ and whose probability distribution is given by
$$\forall x \in X(\Omega), \quad P\left(X^{\prime} = x\right) = \frac{\mathrm{e}^{tx}}{E\left(\mathrm{e}^{tX}\right)} P(X = x)$$
and that there exists a sequence $\left(X_{n}^{\prime}\right)_{n \in \mathbb{N}^{*}}$ of mutually independent random variables defined on $(\Omega, \mathcal{A}, P)$ all following the same distribution as $X^{\prime}$.
b) Show
$$E\left(X^{\prime}\right) = \frac{\varphi_{X}^{\prime}(t)}{\varphi_{X}(t)}, \quad E\left(X^{\prime}\right) > a$$