Changing limits wen integrating with exponential of normally distributed variable

Question

I am working on an exercise which requires calculating the expected value of :

$ E[e^{\mu + \sigma Z} \mathbb{1}_{Z > -d}] $

Calculating the expected value of the $e^{\mu + \sigma Z}$ variable and using the properties of the indicator function, I have reached the point:

$ \frac{1}{\sqrt{2\pi}}e^{\mu + \frac{\sigma ^2}{2}} \int_{-d}^{\infty} e^{-\frac{(z - \sigma)^2}{2}}dz $

However, looking at solutions the following steps is taking $ \sigma$ and moving it into the limit of the integral as follows:

$ \frac{1}{\sqrt{2\pi}}e^{\mu + \frac{\sigma ^2}{2}} \int_{-d - \sigma}^{\infty} e^{-\frac{z^2}{2}}dz $

I am unsure which property has been used here and I was wondering if anyone could point me in the right direction to understanding this switch?

Answer 1

I think I have figured it out, so leaving it here for anyone having a similar question in the future:

If we were to use a change of variable:

let y = $z - \sigma$ then $dy = dz \Longrightarrow \frac{dy}{dz}=1$

Calculating the new limits, we have:

$ z= -d \Longrightarrow y = -d - \sigma $ $ z = \infty \Longrightarrow y = \infty $ This would bring the second result.