Editing Law of total probability (section)

==Continuous case==
The law of total probability extends to the case of conditioning on events generated by continuous random variables. Let <math>(\Omega, \mathcal{F}, P) </math> be a [[probability space]]. Suppose <math> X </math> is a random variable with distribution function <math>F_X</math>, and <math>A</math> an event on <math>(\Omega, \mathcal{F}, P) </math>. Then the law of total probability states 

<math>P(A) = \int_{-\infty}^\infty P(A |X = x) d F_X(x). </math>

If <math>X</math> admits a density function <math>f_X</math>, then the result is 

<math>P(A) = \int_{-\infty}^\infty P(A |X = x) f_X(x) dx. </math>

Moreover, for the specific case where <math>A = \{Y \in B \}</math>, where <math>B</math> is a Borel set, then this yields 

<math>P(Y \in B) = \int_{-\infty}^\infty P(Y \in B |X = x) f_X(x) dx. </math>