Expectations of functions of normal random variables so I am a TA in an intro stats class and I stumbled upon a brain teaser question that even I am not quite sure how to solve. I thought some of you might be able to help. The question is as follows:
You have a normal random variable with mean $\mu$ and standard deviation $\sigma$. The students are asked to calculate the $E[f(X)]$ for three functions (and the question specifies that for the last two, you should express the solution as the standard normal CDF, $\phi$:
a) $f(x)=e^{ax}$ where a is an element of $R$ 
b) $f(x)=(e^{ax}-b)^+$ where a is an element of $R$ and $b>0$$
c) $f(x)=(x-a)^+$ where a is an element of $R$
So, my approach to part (a) was calculating the same as you would for a moment generating function (i.e. just expand it to get $1 + aE(x) + a^2E(x^2)/2! + a^3E(x^3)/3!+...+$) however, I am not sure that that is the correct approach to this question. Then, for the last two questions, I am not sure how to approach them. Any kind of help would be greatly appreciated!
 A: Let us solve (c) and see if you can adapt the ideas below to (a) and (b).


*

*First, to compute $E[f(X)]$ by expanding $f$ into a series is definitely not the first idea one should try. Rather, use the definition (in the density case), that is,
$$
E[f(X)]=\int_\mathbb Rf(x)g(x)\mathrm dx,
$$
where $g$ is the density of the distribution of $X$.

*Second, $X=\sigma Z+\mu$ where $Z$ is standard normal hence it suffices to solve (c) when $X$ has density $\varphi$. Thus, one looks for
$$
E[f(X)]=\int_a^\infty (x-a)\varphi(x)\mathrm dx.
$$

*Third, $\varphi'(x)=-x\varphi(x)$ hence $(x-a)\varphi(x)=-\varphi'(x)-a\varphi(x)$ and
$$
E[f(X)]=\left.-\varphi(x)\right|_a^\infty-a\int_a^\infty\varphi(x)\mathrm dx=\varphi(a)-a+a\Phi(a).
$$

A: For (a): The MGF of the normal distribution can be found here: http://www.cc.gatech.edu/~lebanon/notes/mgf.pdf‎
You basically find the MGF of the standard normal distribution and use that to calculate the MGF of the normal distribution.
For (b), (c) use linearity of expectation.
