Deriving a probability from a moment-generating function

Question

If the MGF of $X$ is $\beta^te^{t^2}$, for some $\beta>0$, find $\Bbb P (X>\text{log}_e(\beta))$.

So far, I can see that the MGF of $X$ is similar to that of a standard normal distribution; what does the $\beta^t$ change though? Do I need to know the PDF of $X$ to compute $\Bbb P (X>\text{log}_e(\beta))$?

Answer 1

Hint: $X$ also has normal distribution. Since $\beta^{t}=e^{ta}$ where $a =\log\, \beta$ $X$ has same distribution as $Y+a$ where $Y$ has MGF $e^{t^{2}}$. Now you can use normal density function to evaluate the probability.