# Derivative of $\int_{-\infty}^{x}(z-x)f(z)\, dz$ with respect to $x$

by Richard Hardy   Last Updated September 11, 2019 09:20 AM - source

I am trying to apply the Leibnitz integral rule to the following problem: $$\frac{d}{dx}\int_{-\infty}^{x}(z-x)f(z)\, dz$$ where $$f(z)$$ is the density function of random variable $$Z$$ (in contrast to the reference above, I use $$z$$ in place of $$t$$). I get \begin{aligned} \frac{d}{dx}\int_{-\infty}^{x}(z-x)f(z)\, dz &= (x-x)f(x)\cdot 1-(-\infty-x)\cdot 0+\int_{-\infty}^{x}(-f(z))\, dz \\ &= 0-\color{red}{(-\infty-x)\cdot 0}-F(x) \end{aligned} where $$F(\cdot)$$ is the cumulative density function of random variable $$Z$$.

Questions:

1. Is the derivation correct?
2. How do I deal with the expression in red?
Tags :