I came across the following claim that puzzles me:
Suppose that $X$ is a random variable with moment generating function $\psi_X$ such that $\psi_X(1) = 2$ and $\psi_X(2) = 4$. Show that $\boldsymbol{\mathrm e}^{X}$ is constant.
First since $t \mapsto \boldsymbol{\mathrm e}^t$ is strictly increasing it should be sufficient to show that $X$ is almost surely constant?
Now from the given information, and the definition of the moment generating function, we have that $$ \int 2 \boldsymbol{\mathrm e}^X dp = \int \boldsymbol{\mathrm e}^{2X} dp . $$
If $X$ has a point mass at $\ln 2$ then the equation is true. Now I wonder if this really is the only possibility?
The equation $$ 2 \boldsymbol{\mathrm e}^t = \boldsymbol{\mathrm e}^{2t} $$ has the single solution $\ln 2$ and \begin{align*} 2 \boldsymbol{\mathrm e}^t > \boldsymbol{\mathrm e}^{2t} &\text{ if } t < \ln 2 \\ 2 \boldsymbol{\mathrm e}^t < \boldsymbol{\mathrm e}^{2t} &\text{ if } \ln 2 < t. \end{align*} What if we take $X$ to have a two point mass, one point to the left and one point to the right of $\ln 2$, of equal probability? Isn't it possible to pick one point to the left where $2 \boldsymbol{\mathrm e}^t - \boldsymbol{\mathrm e}^{2t} = x$ and one point to the right where $\boldsymbol{\mathrm e}^{2t} - 2 \boldsymbol{\mathrm e}^t = x$ and assign $X$ to have equal probability on them?
Thanks in advance!
As per StubbornAtom's comment: From the known formula, that for any random variable $Z$, $$ \mathrm{E}\left[\left(Z- \mathrm WE[Z]\right)^2 \right ] = \mathrm E\left [Z^2 \right] - \mathrm E \left[Z \right]^2, $$ it follows that $$ \mathrm{E}\left [\left(e^X - \mathrm E[e^X] \right)^2\right]= \mathrm E\left[e^{2X}\right] - \mathrm E\left [e^X\right]^2 = \psi(2) - \psi(1)^2 = 0. $$ This means that, almost surely, $e^X$ equals $\mathrm E\left[e^{2X}\right] $. In other words $e^X$ is almost surely constant. This contradicts what I have written in my post. But I'm not sure what is at fault with my original argument.