Leibniz Rule and CDF

Suppose I have a random variable $X$, with support in $\mathbb R$, cdf $F$ and density $f$. Therefore, we have

$P(X\leq \theta)=\int_{-\infty}^\theta f(x)dx=\int_{-\infty}^\theta dF(x)$

Then, by the Leibniz Rule, in the first equality

$\frac{\partial P(X\leq\theta)}{\partial\theta}=f(\theta)$

But in the second equality:

$\int_{-\infty}^\theta dF(x)=\int \mathbb 1(x\leq\theta)dF(x)$, since $\partial\mathbb1(x\leq\theta)/\partial\theta=0$ almost everywhere, then

$\frac{\partial P(X\leq\theta)}{\partial\theta}=0$

• Maybe see this: mathoverflow.net/q/43792 – Sean Roberson Aug 13 '17 at 0:18
• This is the origin of the identity $$\frac{d}{d\theta}1(x\le \theta) = \delta(\theta-x)$$. – spaceisdarkgreen Aug 13 '17 at 0:31
• can you elaborate further please? – julian.marr Aug 13 '17 at 0:33

First, note that this is not specific to probability or the Stieljes integral. You could have asked the question about a plain vanilla integral $$\int_{-\infty}^\theta f(x)dx = \int_{-\infty}^\infty H(\theta-x)f(x)dx$$ where $H$ is the step function. Actually, we can simplify even further and consider derivative with respect to $\theta$ of $$\theta= \int_0^\theta dx = \int_0^\infty H(\theta-x)dx$$ where $\theta > 0.$ Everything about your question still applies here.
Let's go back to the definition of the derivative. We want $$\lim_{h\to0}\frac{f(\theta+h)-f(\theta)}{h} = \lim_{h\to 0} \frac{1}{h}\left(\int_0^\infty H(\theta+h-x)dx - \int_0^\infty H(\theta-x)dx\right).$$ Of course these integrals are just silly ways of writing $\theta+h$ and $\theta$ so they converge and there's no question of us combining them under one and we can write our expression $$\lim_{h\to 0} \int_0^\infty \frac{H(\theta+h-x)-H(\theta-x)}{h}dx.$$
Now we face the all-important question of whether we can bring the limit inside the integral. If we do, we know that we get zero at almost every $x.$ But at the all-important point $x=\theta$ the derivative is undefined. This is where all the action is.
So instead of trying to bring the limit inside, let's look at what the inside function looks like as a function of $x$ for fixed $h.$ It's a rectangle of width $h$ and height $\frac{1}{h}$ with support between $\theta$ and $\theta+h$. Hmm. This is a rectangle of area one that gets arbitrarily tall and thin as $h\to 0.$ So it is an approximation to a Dirac delta. Of course for any $h\ne 0$ the integral is one so the limit is also one, as we knew it had to be.
So, we see the "derivative of the step function" behaves exactly as a delta function. This gives us the distribution identity $$\frac{d}{d\theta} H(\theta-x) = \delta(\theta-x)$$