Conditional Probability Question for A Continuos Random Variable

Question

Suppose $X$ is a continuous random variable with density $p$ with respect to the Lebesgue measure. According to Radon-Nikodym theorem $p$ is measurable itself so it can be seen as a random variable. Question is what is the following expression equal to $$p(X=x_0|p(X=x_0)=3)\overset{?}{=}3$$ Or in a more general but vaguer sense $$p\Big(X=x_0\Big|p(X=x)\ {\rm is\ given\ and\ is\ }f(x)\Big)\overset{?}{=}f(x_0)$$ I have guessed what they need to be equal to --after the question marks-- but I do not have any rigour argument to justify them. So I would really appreciate if you could please guide me how to do it (or prove me wrong).

Answer 1

Let $p_{\small X}(x)$ be the probability density for the continuously distributed random variable $X$.   Then $p_{\small X}(X)$ is a random variable, and you seem to want to talk about the condtional measure $p_{\small X\mid p_{\small X}(X)=3}(x)$.

The nature of the conditional measure, and whether it even exists, entirely depends on the distribution for the random variable.

• If $X$ has a probability density of $3$ in an integrateable interval of uncountable many points, say $A$, then the conditional probability measure $p_{X\mid X\in A}(x)$ will be a probability density function. $$p_{\small X\mid X\in A}(x)=\dfrac{p_{\small X}(x)\mathbf 1_{x\in A}}{\int_A p_{\small X}(s)\mathsf d s}=\dfrac {\mathbf 1_{x\in A}}{\int_A \mathsf d s}$$

• If $X$ has a probability density of $3$ in an interval of finite many points, say $B$, then the conditional probability measure $p_{X\mid X\in B}(x)$ will be a probability mass function. $$p_{\small X\mid X\in A}(x)=\dfrac{p_{\small X}(x)\mathbf 1_{x\in B}}{\sum_{x\in B} p_X(s)}=\dfrac {\mathbf 1_{x\in B}}{\lvert B\rvert}$$

  • For instance, if $X$ has density of $3$ at only one point, say at $x_3$, then the conditional probability that $X=x_3$ when $X=x_3$ will most surely be $1$.

• In other cases it may not even be sensible.