Question involving expectation and standard normal CDF

Question

I am trying to solve this below problem:

Let $Z \sim N(0,1)$, and $c$ be a nonnegative constant. Find $E(max(Z - c, 0))$, in terms of the standard normal CDF $\Phi$ and pdf $\varphi$.

I am struggling with how to get started. Since $Z$ is standard normal, $Z - c$ is standard normal, with the one condition being that it is bounded below by $0$, but not bounded above. The expectation could simply be the integral from $0$ to $\infty$ of $(z-c)$ multiplied by the pdf of the distribution. But, if I integrate this to find the standard normal CDF, the pdf is no longer part of the result, though both are required in this answer. For this reason, I cannot quite figure out the correct approach.

I would greatly appreciate any insights.

Answer 1

It is $\int_c^{\infty} (x-c)\phi (x)\, dx$. To evaluate $\int_c^{\infty} x\phi (x)\, dx$ put note that the derivative of $\phi (x)$ is $-x\phi(x)$. Hence $\int_c^{\infty} x\phi (x)\, dx=\phi (c)$. The answer therefor is $\phi (c) -c[1-\Phi (c)]$.

Answer 2

You might like to investigate the truncated normal distribution

You will find that $E[Z \mid Z \gt c] = \dfrac{\phi(c)}{1- \Phi(c)}$

and so $E[Z-c \mid Z \gt c] = \dfrac{\phi(c)}{1- \Phi(c)}-c$

and thus $E[\max(Z-c,0)] = {\phi(c)}-c(1- \Phi(c))$