Chapter 5, Problem 5.20STPE

For any real number y define by $y^{+}=\begin{array}{cc}y,& \text{if }y\ge 0\\ 0,& \text{if }y<0\end{array}$

Let c be a constant.

a. Show that $E\left[{\left(Z-c\right)}^{+}\right]=\frac{1}{\sqrt{2\pi }}{e}^{-\frac{c^2}{2}}-c\left(1-\varphi \left(x\right)\right)$, when Z is a standard normal random variable.

b. Find $E\left[{\left(Z-c\right)}^{+}\right]$ >when X is normal with mean $\mu$. and variance ${\sigma }^2$