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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 $\mathcal{X}$?

Star
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1 Answers1

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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.

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