# Conditional probability when conditioning on continuous-discrete variables

I am confused on the notion of conditional probability when the conditioning variable is continuous.

Consider the random variables $X,Y$ on the probability space $(\Omega, \mathcal{F}, P)$ with support, respectively, $\mathcal{X}$ and $\mathcal{Y}$. Let $Y$ be a discrete random variable.

(1) If I find $$P(Y=y|X) \hspace{1cm} \text{P-a.s.}$$ and $X$ is continuous does it mean $$P(Y=y|X\in A) \hspace{1cm} \text{\forall A\subseteq \mathcal{X} such that P(X\in A)>0}$$ ?

(2) If I find $$P(Y=y|X) \hspace{1cm} \text{P-a.s.}$$ and $X$ is discrete does it mean $$P(Y=y|X=x) \hspace{1cm} \text{\forall x\in \mathcal{X}}$$ ?

(3) If the answers to (1) and (2) are YES-YES, why in (2) it is sufficient to consider single realisations of $X$ and we can forget non-singleton subsets of $\mathcal{X}$?

• Do you mean $P\{Y\in A\mid X\}$? – d.k.o. May 10 '16 at 19:58
• No, I'm talking about the conditioning. – TEX May 10 '16 at 19:59
• I have edited the question to clarify your point – TEX May 10 '16 at 20:00

By definition, $P\{A\mid X\}$ ($A=\{Y=y\}$ in your case) is a $\sigma(X)$-measurable r.v. satisfying
$$\int_{\{X\in B\}} P\{A\mid X\}dP=P(A\cap \{X\in B\}), \quad \forall B\in\mathcal{B}.$$
($P\{A\mid X\}$ is unique up to null sets). Since $P\{A\mid X\}$ is $\sigma(X)$-measurable, there is a Borel function $\varphi$ s.t. $$\varphi(X(\omega))=P\{A\mid X\}(\omega),$$ and $\varphi(x)$ is denoted by $P\{A\mid X=x\}$. This construction works in both cases (for continuous and discrete $X$) and if you know $\varphi(x)$, you can find $P\{A\mid X\}$ and vice versa.
• $P(A|X=x)$ does not make sense if $X$ is continuous because $P(X=x)=0$ – TEX May 11 '16 at 10:21
• Unless, we define $P(A|X=x)$ when $X$ is continuous using the pdf of $X$. This is the point is confusing me I guess – TEX May 11 '16 at 10:30
• The meaning of conditional probability/expectation is the same for both discrete and continuous $X$'s - it's a function of $X$ and this function is denoted by $P\{A\mid X=x\}$. When $B=\{X=x\}$ is not a null set $P\{A\mid B\}$ has the ordinary meaning: probability of $A$ given $B$. – d.k.o. May 11 '16 at 19:07