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$1/(1+t)$ $($ $ -1<t<1)$ seems to be a legitimate moment generating function (MGF) since its value is $1$ where $t=0$, but is there a way to find the probability density function of $X$?

I know $1/(1-t)$ is the MGF of the exponential distribution, but I have no idea with $1/(1+t)$. Any help would be appreciated.

Richard
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1 Answers1

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Hint: This is $M_X(-t)$, where $X$ is Exponential with parameter $1$. Then use the general fact that $M_X(at) = M_{aX}(t)$ for any constant $a$ (i.e. $M_X(at)$ is the MGF of $aX$) to conclude what random variable would have that MGF.

Minus One-Twelfth
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