I stumbled upon this homework question and I simply cannot wrap my head around it:
As far as I understood it, I would start off with the following:
But I get completely stuck from here on out. Perhaps my first move is already wrong. Could anyone please help?
Thank you very much in advance!
You should begin with
$\int\limits_{-\infty}^\infty e^{mx} I_{X \leq d} f(x) dx$
where $I$ is the indicator, $f$ is the density function. This then becomes
$\int\limits_{-\infty}^d e^{mx} \frac{1}{\sqrt{2 \pi}} e^{-x^2/2} dx$
The main trick you'll want to use is $e^{mx} e^{-x^2 / 2} = e^{-x^2/2 + mx}$. Then, you'll want to complete the square.