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!
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
$$
E[f(X)]=\left.-\varphi(x)\right|_a^\infty-a\int_a^\infty\varphi(x)\mathrm dx=\varphi(a)-a+a\Phi(a).
$$
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.