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Conditional cumulative distribution. |
More about conditional CDFs. |
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Given a random variable <math>X</math> and a [[event (probability theory)|random event]] <math>A</math>, the conditional cumulative distribution of <math>X</math> given <math>A</math> is defined by<ref name=KunIlPark>{{cite book | author=Park,Kun Il| title=Fundamentals of Probability and Stochastic Processes with Applications to Communications| publisher=Springer | year=2018 | isbn=978-3-319-68074-3}}</ref>{{rp|p. 97}}
:<math>F_{X|A}(x) \triangleq \frac{P(X \leq x \cap A)}{P(A)}</math>
for <math>P(A) > 0</math>.
If another random variable is denoted by <math>Y</math>, it is possible to condition on the event <math>\{Y \leq y \}</math>. This yields
:<math>F_{X|Y \leq y}(x|y) = \frac{P(X \leq x \cap Y \leq y)}{P(Y \leq y)}</math>
which can be written as
:<math>F_{X|Y \leq y}(x|y) = \frac{F_{X,Y}(x,y)}{F_Y(y)}</math>
where <math>F_{X,Y}(x,y)</math> denotes the joint cumulative distribution function of <math>X</math> and <math>Y</math> and <math>F_Y(y)</math> is the cumulative distribution function of <math>Y</math>.
==Conditional discrete distributions==
For [[discrete random variable]]s, the [[conditional probability]] mass function of <math>Y</math> given the occurrence of the value <math>x</math> of <math>X</math> can be written according to its definition as:
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==Conditional continuous distributions==
Similarly for [[continuous random variable]]s, the conditional [[probability density function]] of <math>Y</math> given the occurrence of the value <math>x</math> of <math>X</math> can be written as<ref name=KunIlPark/>{{rp|p. 99}}
{{Equation box 1
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