Conditional probability distribution: Difference between revisions

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==Conditional discrete distributions==

For [[discrete random variable]]s, the conditional [[probability mass function]] of <math>Y</math> given <math>X=x</math> can be written according to its definition as:

{{Equation box 1

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Then the unconditional probability that <math>X=1</math> is 3/6 = 1/2 (since there are six possible rolls of the dice, of which three are even), whereas the probability that <math>X=1</math> conditional on <math>Y=1</math> is 1/3 (since there are three possible [[prime number]] rolls—2, 3, and 5—of which one is even).

==Conditional continuous distributions==

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==Relation to independence==

Random variables <math>X</math>, <math>Y</math> are [[Statistical independence|independent]] [[if and only if]] the conditional distribution of <math>Y</math> given <math>X</math> is, for all possible realizations of <math>X</math>, equal to the unconditional distribution of <math>Y</math>. For discrete random variables this means <math>P(Y=y|X=x) = P(Y=y)</math> for all possible <math>y</math> and <math>x</math> with <math>P(X=x)>0</math>. For continuous random variables <math>X</math> and <math>Y</math>, having a [[joint density function]], it means <math>f_Y(y|X=x) = f_Y(y)</math> for all possible <math>y</math> and <math>x</math> with <math>f_X(x)>0</math>.

==Properties==

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==Measure-theoretic formulation==

Let <math>(\Omega, \mathcal{F}, P)</math> be a [[probability space]], <math>\mathcal{G} \subseteq \mathcal{F}</math> a <math>\sigma</math>-field in <math>\mathcal{F}</math>. Given <math>A\in \mathcal{F}</math>, the [[Radon-NikodymRadon–Nikodym theorem]] implies that there is{{sfnp|Billingsley|1995|p=430}} a <math>\mathcal{G}</math>-measurable random variable <math>P(A\mid\mathcal{G}):\Omega\to \mathbb{R}</math>, called the [[conditional probability]], such that<math display="block">\int_G P(A\mid\mathcal{G})(\omega) dP(\omega)=P(A\cap G)</math>for every <math>G\in \mathcal{G}</math>, and such a random variable is uniquely defined up to sets of probability zero. A conditional probability is called [[Regular conditional probability|'''regular''']] if <math> \operatorname{P}(\cdot\mid\mathcal{G})(\omega) </math> is a [[probability measure]] on <math>(\Omega, \mathcal{F})</math> for all <math>\omega \in \Omega</math> a.e.

Special cases:

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An expectation of a random variable with respect to a regular conditional probability is equal to its conditional expectation.

=== Interpretation of conditioning on a Sigma Field ===

Consider the probability space <math>(\Omega, \mathcal{F}, \mathbb{P})</math>

and a sub-sigma field <math>\mathcal{A} \subset \mathcal{F}</math>.

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Also recall that an event <math>B</math> is independent of a sub-sigma field <math>\mathcal{A}</math> if <math>\mathbb{P}(B | A) = \mathbb{P}(B)</math> for all <math>A \in \mathcal{A}</math>. It is incorrect to conclude in general that the information in <math>\mathcal{A}</math> does not tell us anything about the probability of event <math>B</math> occurring. This can be shown with a counter-example:

Consider a probability space on the [[unit interval]], <math>\Omega = [0, 1]</math>. Let <math>\mathcal{G}</math> be the sigma-field of all countable sets and sets whose complement is countable. So each set in <math>\mathcal{G}</math> has measure <math>0</math> or <math>1</math> and so is independent of each event in <math>\mathcal{F}</math>. However, notice that <math>\mathcal{G}</math> also contains all the singleton events in <math>\mathcal{F}</math> (those sets which contain only a single <math>\omega \in \Omega</math>). So knowing which of the events in <math>\mathcal{G}</math> occurred is equivalent to knowing exactly which <math>\omega \in \Omega</math> occurred! So in one sense, <math>\mathcal{G}</math> contains no information about <math>\mathcal{F}</math> (it is independent of it), and in another sense it contains all the information in <math>\mathcal{F}</math>.{{sfnp|Billingsley|2012}}{{Page needed|date=May 2025}}

== See also ==

* [[Conditioning (probability)]]

* [[Conditional probability]]

* [[Regular conditional probability]]

* [[Bayes' theorem]]

== References ==

=== Citations ===

{{Reflist}}

=== Sources ===

{{refbegin}}

* {{cite book |last= Billingsley |first= Patrick |date= 1995 |title= Probability and Measure |edition= 3rd |publisher= John Wiley and Sons |___location= New York |isbn= 0-471-00710-2 |author-link= Patrick Billingsley |url= https://books.google.com/books?id=a3gavZbxyJcC }}

* {{cite book |last= Billingsley |first= Patrick |date= 2012 |title= Probability and Measure |edition= Anniversary |publisher= Wiley |___location= Hoboken, New Jersey |isbn= 978-1-118-12237-2 |url= https://www.amazon.com/Probability-Measure-Patrick-Billingsley/dp/1118122372 }}

* {{cite book |last= Park |first= Kun Il |date= 2018 |title= Fundamentals of Probability and Stochastic Processes with Applications to Communications |publisher= Springer |isbn= 978-3-319-68074-3}}

* {{cite book |last= Ross |first= Sheldon M. |date= 1993 |title=Introduction to Probability Models |edition= 5th |___location= San Diego |publisher= Academic Press |isbn=0-12-598455-3 |author-link= Sheldon M. Ross }}