I´m working on trying to approach the value of $E\left[ \dfrac{e^x}{x+1} \right]$. Where $x$ is an exponential random variable. All that data I have to work with is a gamma random variable with parameters $3/2$ and $1$. I also know that $x$ has mean $1$.
I think that I could get some relations by knowing that if I sum k exponential random variables I would get a gamma random variable. But I´m a bit confused on how I could apply this to the computation of the Expected Value. Can anyone suggest any valid process to compute the above value using the fact that I know the Gamma Random Variable??
EDIT:
POSSIBLE ANSWER: I have received several comments about the divergence of the integral, however I think I can use this estimator to approach the value:
Where $T$ is the expected value of the gamma random variable and $Si$ the i-th value generated by the gamma distribution
Thank you
I'm not sure whether you mean $E\left[ \dfrac{e^x}{x+1} \right]$ or $E\left[ \dfrac{e^x}{x} + 1\right]$. But neither of these exists (the improper integrals diverge). Your gamma random variable is irrelevant to this question.