I am working on a problem that I always believed I understood how to do but I cannot seem to get the correct answer. I am given the pdf $f(x) = .5e^{-|x|}$ for x from negative infinity to infinity. I am supposed to be showing that the Moment generating function is $M(t)=1/{(1-t^2)}$ for $|t|<1$
So i already know what the answer is supposed to be but I cannot seem to get there so I am wondering if I am misunderstanding something
The moment Generating function is $E(e^{tX})$ so it would be $\int e^{tx}.5e^{|x|}$ correct?
I simplied this to $.5\int e^{tx-|x|}$ Then took the antiderivative
I have calculated this out to be $.5(1/t-1)e^{tx-|x|}$
I have never been great with derivates and integral of $e$ so if this is where all of my problems lie that would not be surprised but if it is wrong if someone could walk through it that would be amazing.
I am slightly confused on if the limits of the integral are negative infinity to infinity or from 0 to 1 since that is the limit of t. Either way I do not get the correct answer
I am either having issues with my integral or just my general understanding, any insight would be greatly appreciated
You're taking the indefinite integral, while it should be the definite integral from $-\infty$ to $\infty$.
Once you set the integral limits, your only problem is $|x|$, which can be solved by breaking up the integral when $x<0$ and $x>0$.