How to find $E(X|X\leq c)$?

Question

I have a random variable $X$ and I am interested in finding the following $$E[X|X\leq c]$$ where $c$ is some positive constant. Using my preliminary knowledge I can write $$E[X|X\leq c]=\int_{-\infty}^c xp_X(x|x\leq c)dx$$ I do not know how to get $p_X(x|x\leq c)$? How to get $p_X(x|x\leq c)$? Any help in this regard will be much appreciated. Thanks in advance.

Answer 1

Note that $$E[X|X\leq c]=\int_{-\infty}^c xp_X(x|x\leq c)dx$$ is the same as $$E[X|X\leq c]=\frac{\int_{-\infty}^c xp_X(x)dx}{P(X\leq c)}$$ by Bayes' formula, below, for a continuous random variable $X$ and assuming $\{X\leq c\}$ is an event of positive measure,

$$\begin{align}p_X(x\mid X\leq c) & =\lim_{h\to 0}\mathsf P(X \in [x,x+h)\mid X \in (-\infty,c])/h \\ &= \lim_{h\to 0} \frac{\mathsf P(X \in [x,x+h),X \in (-\infty,c])}{h~\mathsf P(X \in (-\infty,c])}\\ &= \lim_{h\to 0} \frac{\mathsf P(X \in [x,\min\{x+h,c\}),x \in (-\infty,c))}{h~\mathsf P(X \in (-\infty,c])} \\ &= \frac{p_X(x)~\mathbf 1_{x\in(-\infty ;c]}}{\mathsf P(X \in (-\infty,c])}\end{align}$$