I need to calculate the expected value of a modified normal distributed variable but i'm struggling. So maybe someone can help me.
Suppose we've got a normal distributed variable $X \sim \mathcal{N}(0,\sigma^2)$. Using this variable $X$ we define $Y$,
$Y:=-\left(a + \frac{X}{\sigma}\right), a\le 1 \in \mathbb{R}$.
Now i need help calculating the expected value of $Y^+$.
One can find similar terms in financial mathematics, e.g. the expected value of a put option $P = \left(K-S\right)^+$ is given by
$\mathbb{E}[P] = \int^K_0 \left(K-S\right)f\left(S\right)dS$,
with $f(S)$ being the probability density function of $S$.
But i just can't apply this result in order to find $\mathbb{E}[Y^+]$ and validate that it is positive for $a<1$. I would really appreciate any helpful comments. Thanks in advance!