expected value, random variable, piecewise function

Question

I hope that you can help me with the following problem:

Let $X \sim \mathcal N\left(\mu, \sigma^2\right)$ be a random variable. Define a new random variable $Y$ as:

$$ Y = g(X) = \begin{cases} a(z+X) & \text{if } (z+X) \leq k \\ ak + b(z+X-k) & \text{if } (z+X) > k, \end{cases} $$ where $a$, $b$, $k$, and $z$ are scalars. I want to calculate the expected value of $E(Y)$. Unfortunately, I don't know how to proceed. I guess I should apply the transformation theorem:

$$ E(Y) = \int _{-\infty}^{\infty} g(x)dF_x(x) $$

But I don't know hot to apply it to $g(X)$.Thank you for your help!

Answer 1

For a random variable $X$ with finite first and second moments (i.e. expectation and variance exist) it holds that $\forall c\in\mathbb R$, $\mathbb E[cX]=c\mathbb E[X]$.

In your case it is going to be pretty straight forward assuming that $a$,$b$,$k$ and $z$ are scalars and that the expectation of a normal distribution is the mean $\mu$:

$$ \mathbb E[a(z+X)] = \mathbb E[az + aX] = \mathbb E[az] + \mathbb E[aX] = az + a\mathbb E[X] = az + aμ$$ for $k\leqslant z+X$.

Now just follow the same line of thought for the other equation.