I am trying to find a formula for the conditional probability density below:
$f_X(x| X < M)$
where $M$ is a constant and $X$ is a continuous random variable.
I want to relate the formula of the conditional probability density with the probability distribution of $X$ (whether it is density function $f_X(x)$ or cumulative function $F_X(x)$). But I can't do that, could you please suggest me a solution ?
Thank you very much for your help!
Best Answer
When $x < M$ you have $f_X(x)=f_X(x\mid X < M) \,\mathbb P(X <M)$, so
$$f_X(x\mid X < M) =\dfrac{f_X(x)}{\mathbb P(X <M)} \text{ if } x <M$$ and $0$ otherwise.
If $X$ has a continuous distribution then $\mathbb P(X <M) = F_X(M) = \int\limits_{-\infty}^M f_X(x) \, dx$