Integral of a Conditional Probability

Question

I'm trying to calculate the derivative of an integral of a conditional probability but a bit unsure how to go about it.

Specifically, I have:

$f = \int_{-\infty}^{x'} P(x \mid c) dx$

And I want to calculate $f'(x)$

Any suggestions much appreciated!

Answer 1

It's a well-know property that if $$f(x) = \int_{a}^x g(t)dt,$$ for some integrable function $g$ and a fixed value $a$ (can be $-\infty$), then $$f'(x) = g(x).$$ Therefore here the answer is just $f'(x) = P(x|c)$.

Another approach is to interpret $f$ as the conditional cumulative distribution function $f(x) = P(X\leq x|c)$ (I think you mean that $P(x|c)$ is the conditional p.d.f of X at point x). Then by definition the derivative of $f$ is the conditional p.d.f at $x$, i.e., $f'(x) = P(x|c)$.