Random Variable Property $E[e^X] = e^{E[X]}$

The question deals with a real-valued random variable $X$ that satisfies two conditions: $E[e^X] < \infty$ and $E[e^X] = e^{E[X]}$. We need to determine which of the given statements is correct based on these conditions.

Jensen's Inequality and Convex Functions

This problem involves the concept of Jensen's inequality. Jensen's inequality states that for a convex function $\phi$ and a random variable $X$, $E[\phi(X)] \ge \phi(E[X])$. The exponential function, $f(x) = e^x$, is a strictly convex function.

For a strictly convex function $\phi$, the equality $E[\phi(X)] = \phi(E[X])$ holds if and only if $X$ is a constant random variable almost surely. In this case, the function is $f(x) = e^x$, which is strictly convex. The given condition is $E[e^X] = e^{E[X]}$.

Therefore, the condition $E[e^X] = e^{E[X]}$ implies that the random variable $X$ must be a constant almost surely. Let this constant be $c$, so $X = c$ almost surely.

The condition $E[e^X] < \infty$ ensures that the expectation $E[e^X]$ exists and is finite. If $X=c$, then $E[e^X] = E[e^c] = e^c$. Since $c$ is a real number, $e^c$ is a finite positive number, so $e^c < \infty$ is always true for any real constant $c$.

Evaluating the Options based on X being a Constant

Since $X=c$ almost surely, where $c$ is some real constant, let's examine each option:

1. $P(X \ge a) \ge e^{E[X]-a}$ for all $a \in \mathbb{R}$

   If $X=c$, then $E[X]=c$. The inequality becomes $P(c \ge a) \ge e^{c-a}$.

   • If $a \le c$, $P(c \ge a) = 1$. The inequality is $1 \ge e^{c-a}$. This is true if $e^{c-a} \le 1$, which means $c-a \le 0$, or $a \ge c$. This contradicts $a \le c$ unless $a=c$. So, for $a < c$, $1 \ge e^{c-a}$ is not always true (e.g., if $c-a = 1$, $1 \ge e^1 \approx 2.718$ is false).
   • If $a > c$, $P(c \ge a) = 0$. The inequality is $0 \ge e^{c-a}$. Since $e^{c-a}$ is always positive, $0 \ge e^{c-a}$ is false.

   Therefore, this option is not correct for all $a \in \mathbb{R}$.

2. $E[X^3] = (E[X])^3$

   If $X=c$, then $E[X] = E[c] = c$.

   $E[X^3] = E[c^3] = c^3$ (since $c^3$ is a constant).

   $(E[X])^3 = (c)^3 = c^3$.

   Thus, $E[X^3] = c^3 = (E[X])^3$. This statement holds true if $X$ is a constant.

3. $Var(X) \ne 0$

   If $X=c$, then the variance of $X$ is $Var(X) = Var(c) = 0$. A constant random variable has zero variance.

   The statement $Var(X) \ne 0$ is false if $X$ is a constant.

4. $X \ge 0$ almost surely

   If $X=c$ almost surely, this means $c \ge 0$. The condition $E[e^X] = e^{E[X]}$ implies $X$ is a constant $c$. This constant $c$ can be any real number (e.g., $c=-5$). The given conditions do not restrict the sign of the constant $c$. For example, if $X=-1$, $E[e^X] = e^{-1}$ and $e^{E[X]} = e^{-1}$, so $E[e^X] = e^{E[X]}$ holds, but $X \ge 0$ is false.

   This statement is not necessarily correct.

Based on the analysis, only the statement $E[X^3] = (E[X])^3$ is always true when $X$ is a constant random variable, which is implied by the condition $E[e^X] = e^{E[X]}$.