Editing Conditional expectation (section)

=== Continuous random variables ===
Let <math>X</math> and <math>Y</math> be [[continuous random variable]]s with joint density
<math>f_{X,Y}(x,y),</math>
<math>Y</math>'s density
<math>f_{Y}(y),</math>
and conditional density <math>\textstyle f_{X\mid Y}(x\mid y) = \frac{ f_{X,Y}(x,y) }{f_{Y}(y)}</math> of <math>X</math> given the event <math>Y=y.</math>
The conditional expectation of <math>X</math> given <math>Y=y</math> is
:<math>
\begin{aligned}
\operatorname{E} (X \mid Y=y) &=  \int_{-\infty}^\infty x f_{X\mid Y}(x\mid y) \, \mathrm{d}x \\
&= \frac{1}{f_Y(y)}\int_{-\infty}^\infty x f_{X,Y}(x,y) \, \mathrm{d}x.
\end{aligned}
</math>
When the denominator is zero, the expression is undefined.

Conditioning on a continuous random variable is not the same as conditioning on the event <math>\{ Y = y \}</math> as it was in the discrete case. For a discussion, see [[Conditional probability#Conditioning on an event of probability zero|Conditioning on an event of probability zero]]. Not respecting this distinction can lead to contradictory conclusions as illustrated by the [[Borel-Kolmogorov paradox]].