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 $\mathcal{X}$?

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

  • $\begingroup$ Thanks. Could you add some comments more related to my questions? $\endgroup$ – TEX May 11 '16 at 10:18
  • $\begingroup$ $P(A|X=x)$ does not make sense if $X$ is continuous because $P(X=x)=0$ $\endgroup$ – TEX May 11 '16 at 10:21
  • $\begingroup$ 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 $\endgroup$ – TEX May 11 '16 at 10:30
  • $\begingroup$ 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 $B$. $\endgroup$ – d.k.o. May 11 '16 at 19:07

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