Why can't I get the conditional probability from the joint probability by simply substituting a value of the random variable in the joint probability?

Question

For two continuous random variables $X$ and $Y$, why can't I get the conditional probability $f(Y|X=x)$ from the joint probability $f(X,Y)$ by simply substituting $X=x$ in the joint probability $f(X,Y)$? Alternatively Wikipedia states "in some cases the conditional probabilities may be expressed as functions containing the unspecified value $x$ of $X$ as a parameter". In such a case how is the conditional probability different from the joint probability?

Answer 1

In such a case how is the conditional probability different from the joint probability?

The difference is the normalizing factor. In the case of discrete random variables ( with probability masses ), then for all supported values :

$$\mathsf P(Y=y\mid X=x)={\dfrac1{\mathsf P(X=x)}\,\mathsf P(X=x, Y=y)}$$

Likewise for continuous random variables with well defined pdf.

$$f_{Y\mid X}(y\mid x)={\dfrac1{f_X(x)}\,f_{X,Y}(x, y)}$$