# Conditional probability distribution

A conditional probability distribution is the probability distribution of a random variable or random element when some other information is treated as known, written for example as the distribution of \( X \) given that \( Y = y \) or given a sub-\( \sigma \)-algebra \( \mathcal{G} \) of the underlying probability space. In elementary settings it is computed from a joint distribution divided by a marginal; in full measure-theoretic probability it is formalized as a regular conditional distribution, a probability kernel that assigns to each conditioning outcome an entire probability measure.<sup>[1](https://www.statlect.com/fundamentals-of-probability/conditional-probability-distributions)</sup><sup> • </sup><sup>[2](https://encyclopediaofmath.org/wiki/Conditional_density)</sup>

| Key fact | Detail |
|---|---|
| Discrete case | The conditional pmf of \( X \) given \( Y = y \) is the joint pmf divided by the marginal pmf of \( Y \), the usual conditional-probability formula applied pointwise<sup>[1](https://www.statlect.com/fundamentals-of-probability/conditional-probability-distributions)</sup> |
| Continuous case | The conditional density is \( f_{X}(x \mid Y = y) = f_{X,Y}(x,y) / f_{Y}(y) \) for values of \( y \) with \( f_{Y}(y) \neq 0 \)<sup>[2](https://encyclopediaofmath.org/wiki/Conditional_density)</sup> |
| Regular version | A regular conditional distribution is a kernel \( \mu(\omega, A) \) that is a probability measure in \( A \) for \( \mathbb{P} \)-almost every \( \omega \) and equals \( \mathbb{P}(X \in A \mid \mathcal{G})(\omega) \)<sup>[3](https://webhost.bridgew.edu/j1pike/Prob%20Theory%20II%20notes.pdf)</sup> |
| Existence | Existence for all random elements is a property of the target space: it always holds on \( (\mathbb{R}, \mathcal{B}(\mathbb{R})) \) and on standard Borel spaces, including Polish spaces with their Borel \( \sigma \)-algebra; counterexamples exist in general<sup>[4](https://www.math.ucdavis.edu/~gravner/MAT235B/assignments/MAT235B_lec1.pdf)</sup><sup> • </sup><sup>[5](https://androma.org/theorems/17345)</sup> |
| Uniqueness | For each measurable set in the target space separately, any two regular conditional distributions agree off a \( \mathbb{P} \)-null set in \( \mathcal{G} \); a single common null set in \( \mathcal{G} \) works when the target sigma-algebra is countably generated<sup>[5](https://androma.org/theorems/17345)</sup><sup> • </sup><sup>[16](https://galton.uchicago.edu/~lalley/Courses/383/ConditionalExpectation.pdf)</sup> |
| Zero-probability conditioning | When \( \mathbb{P}(Y = y) = 0 \), the conditional distribution of \( X \) given \( Y = y \) cannot in general be derived unambiguously, the Borel–Kolmogorov paradox<sup>[1](https://www.statlect.com/fundamentals-of-probability/conditional-probability-distributions)</sup> |
| Main use | Conditional expectations of functions of \( X \) given \( \mathcal{G} \) can be computed as integrals against the regular conditional distribution, analogous to the change-of-variables formula<sup>[3](https://webhost.bridgew.edu/j1pike/Prob%20Theory%20II%20notes.pdf)</sup> |

## Elementary conditional distributions

For discrete random vectors, the conditional distribution is defined pointwise from the joint probability mass function. If \( p_{X,Y}(x,y) \) is the joint pmf and \( p_{Y}(y) \) the marginal pmf of \( Y \), obtained by marginalization when it is not known directly, then the conditional pmf of \( X \) given \( Y = y \) is the ratio of the two, provided \( p_{Y}(y) > 0 \). This is the ordinary formula for conditional probability, joint probability divided by marginal probability, applied to each value of \( x \).<sup>[1](https://www.statlect.com/fundamentals-of-probability/conditional-probability-distributions)</sup>

For continuous random vectors the analogous object is the conditional probability density function. Writing \( f_{X,Y} \) for the joint density and \( f_{Y} \) for the marginal density of \( Y \), the conditional density of \( X \) at \( x \) given \( Y = y \) is

\[ f_{X}(x \mid Y = y) = \frac{1}{f_{Y}(y)} f_{X,Y}(x,y) \]

for every \( y \) with \( f_{Y}(y) \neq 0 \).<sup>[2](https://encyclopediaofmath.org/wiki/Conditional_density)</sup> Equivalently, a function is the conditional distribution function of \( X \) given \( Y = y \) when its values give the conditional probability that \( X \) lies below each threshold.<sup>[6](https://new.statlect.com/fundamentals-of-probability/conditional-probability-distributions)</sup>

**The zero-probability problem.** The ratio formulas presuppose that the conditioning event has positive probability. When \( \mathbb{P}(Y = y) = 0 \), as happens for every exact value of a continuous variable, it is in general not possible to derive the conditional distribution of \( X \) given \( Y = y \) unambiguously. This impossibility is known as the Borel–Kolmogorov paradox; StatLect notes that it is seldom relevant in applications, but it motivates the more careful measure-theoretic treatment below.<sup>[1](https://www.statlect.com/fundamentals-of-probability/conditional-probability-distributions)</sup>

## Conditioning on a sigma-algebra

In measure-theoretic probability, the information available is encoded as a sub-\( \sigma \)-algebra \( \mathcal{G} \) of the \( \sigma \)-algebra \( \mathcal{F} \) on the sample space \( \Omega \). For a random element \( X \) taking values in a measurable space \( (S, \mathcal{S}) \), the conditional probability \( \mathbb{P}(X \in A \mid \mathcal{G}) \) is defined for each fixed set \( A \in \mathcal{S} \) as a version of the conditional expectation of the indicator of \( \{X \in A\} \). The difficulty is that each set \( A \) comes with its own version, defined only up to a null set, and the null sets can depend on \( A \), so the pointwise-defined map \( A \mapsto \mathbb{P}(X \in A \mid \mathcal{G})(\omega) \) need not be countably additive for any single \( \omega \).<sup>[7](https://math.iisc.ac.in/~gadgil/proofs-and-programs-2023/doc/Mathlib/Probability/Kernel/CondDistrib.html)</sup>

A version of the conditional probabilities for which they do form a measure is called a regular conditional probability, and every regular conditional probability is a probability kernel.<sup>[8](https://www.stat.cmu.edu/~cshalizi/754/notes/lecture-03.pdf)</sup> Formally, a function \( \mu : \Omega \times \mathcal{S} \to [0,1] \) is a regular conditional distribution of \( X \) given \( \mathcal{G} \) when two conditions hold: for \( \mathbb{P} \)-almost every \( \omega \), the map \( A \mapsto \mu(\omega, A) \) is a probability measure on \( (S, \mathcal{S}) \), and for each \( A \in \mathcal{S} \) the function \( \omega \mapsto \mu(\omega, A) \) is a version of \( \mathbb{P}(X \in A \mid \mathcal{G}) \).<sup>[3](https://webhost.bridgew.edu/j1pike/Prob%20Theory%20II%20notes.pdf)</sup> When the target space is the original sample space itself, \( (S, \mathcal{S}) = (\Omega, \mathcal{F}) \) with \( X(\omega) = \omega \), the object is called a regular conditional probability.<sup>[3](https://webhost.bridgew.edu/j1pike/Prob%20Theory%20II%20notes.pdf)</sup><sup> • </sup><sup>[9](https://web.ma.utexas.edu/users/gordanz/notes/theory_of_probability_I.pdf)</sup>

The regular conditional distribution is the conditional analog of the pushforward measure \( \mathbb{P} \circ X^{-1} \) that an ordinary random variable induces on its value space: it describes what the distribution of \( X \) looks like from within each fiber of the conditioning information.<sup>[3](https://webhost.bridgew.edu/j1pike/Prob%20Theory%20II%20notes.pdf)</sup> The nLab describes the same object as a probability measure on the target space parametrized in a measurable way by the conditioning variable.<sup>[10](https://ncatlab.org/nlab/show/Markov%2Bkernel)</sup>

## Existence and uniqueness

Whether a regular conditional distribution exists for every random element is a property of the target space \( (S, \mathcal{S}) \), not of the probability space alone. Existence always holds when \( (S, \mathcal{S}) = (\mathbb{R}, \mathcal{B}(\mathbb{R})) \), and more generally whenever \( \mathcal{S} \) is generated by a countable collection of sets; such spaces are called Borel spaces. Every complete separable metric space, a [Polish space](https://www.edgepedia.org/polish-space), with its Borel \( \sigma \)-algebra is a Borel space, since its \( \sigma \)-algebra is generated by the open balls of rational radii around the points of a countable dense subset. Counterexamples exist outside this class, but for practical purposes regular conditional distributions always exist.<sup>[4](https://www.math.ucdavis.edu/~gravner/MAT235B/assignments/MAT235B_lec1.pdf)</sup>

The standard Borel space assumption gives a precise existence theorem: if \( (\Omega, \mathcal{F}) \) and \( (E, \mathcal{E}) \) are standard Borel spaces, \( \mathbb{P} \) is a probability measure on \( \Omega \), \( \mathcal{G} \subset \mathcal{F} \) is countably generated, and \( Y : \Omega \to E \) is measurable, then a regular conditional distribution \( K \) of \( Y \) given \( \mathcal{G} \) exists. Moreover, if \( K_1 \) and \( K_2 \) are two such kernels, then for every \( B \in \mathcal{E} \) there is a set \( N \in \mathcal{G} \) with \( \mathbb{P}(N) = 0 \) such that \( K_1(\omega, B) = K_2(\omega, B) \) for every \( \omega \notin N \), so the two kernels agree almost surely for each measurable target set separately; when \( \mathcal{E} \) is countably generated, a single null set works for all \( B \) simultaneously, and the kernel is unique up to a null set.<sup>[5](https://androma.org/theorems/17345)</sup><sup> • </sup><sup>[16](https://galton.uchicago.edu/~lalley/Courses/383/ConditionalExpectation.pdf)</sup>

The same uniqueness-up-to-null-sets appears in the formalization of the concept in the Lean mathematical library Mathlib, where `condDistrib Y X μ` is defined as a [Markov kernel](https://www.edgepedia.org/markov-kernel) associated with the conditional expectation of \( Y \) given \( X \), requiring \( \Omega \) to be a standard Borel space and \( \mu \) a finite measure; for almost all points the kernel evaluated at \( X a \) and a measurable set \( s \) equals the conditional expectation of the indicator of \( Y^{-1}(s) \).<sup>[7](https://math.iisc.ac.in/~gadgil/proofs-and-programs-2023/doc/Mathlib/Probability/Kernel/CondDistrib.html)</sup>

## Relation to disintegration

Regular conditional distributions are the probability-space instance of disintegration of measures. Given a \( \sigma \)-finite measure \( \mu \) on a product \( X \times T \), a disintegration consists of a \( \sigma \)-finite measure \( \nu \) on \( X \) and a Markov kernel \( \rho \) from \( X \) to \( T \) such that \( \mu(B \times A) = \int_{B} \rho_{x}(A) \, \nu(dx) \) for all measurable \( B, A \); the inverse operation is called mixing. A disintegration exists when \( (T, \mathcal{T}) \) is Polish Borel, the measure \( \nu \) is unique up to equivalence, and the kernel \( \rho \) is unique up to a measurable renormalization almost everywhere.<sup>[11](https://encyclopediaofmath.org/wiki/Stochastic_point_process)</sup> On a [standard probability space](https://www.edgepedia.org/standard-probability-space), the conditional measures \( P_{\omega'} \) on the fibers \( \{\omega : \alpha(\omega) = \omega'\} \) of a measurable map \( \alpha \) play exactly this role; for the map \( (x,y) \mapsto x \) on a square, the conditional distribution of \( (x,y) \) given \( x \) is one-dimensional [Lebesgue measure](https://www.edgepedia.org/lebesgue-measure) on the vertical fiber \( \{x\} \times (0,1) \).<sup>[12](https://encyclopediaofmath.org/wiki/Standard_probability_space)</sup>

## Uses

The practical value of a regular conditional distribution is that it turns conditioning into ordinary integration. If \( \mu \) is a regular conditional distribution for \( X \) given \( \mathcal{G} \), then the conditional expectation of any measurable function \( g \) of \( X \) with finite expectation is \( \mathbb{E}[g(X) \mid \mathcal{G}](\omega) = \int_{S} g(x) \, \mu(\omega, dx) \), just as the ordinary change-of-variables formula computes expectations against a distribution.<sup>[3](https://webhost.bridgew.edu/j1pike/Prob%20Theory%20II%20notes.pdf)</sup> In the concrete case of a pair \( (X, Y) \) with joint density \( f \) and \( \mathcal{G} = \sigma(Y) \), the regular conditional distribution is the random measure with density \( f(x, Y) / f_{Y}(Y) \) with respect to Lebesgue measure, recovering the elementary conditional density formula.<sup>[4](https://www.math.ucdavis.edu/~gravner/MAT235B/assignments/MAT235B_lec1.pdf)</sup>

Conditional distributions also carry structural theorems. For an exchangeable sequence of random elements in a standard Borel space, the sequence is conditionally independent and identically distributed given the exchangeable \( \sigma \)-algebra if and only if each random element admits a regular conditional distribution given that \( \sigma \)-algebra, a generalization of the de Finetti–Hewitt–Savage theorem.<sup>[13](https://arxiv.org/abs/2312.16349)</sup> In stochastic processes, the conditional distribution of the future given the present defines the transition law of a Markov process, in which the distribution of \( X(t_1) \) given the whole past depends only on \( X(t_0) \).<sup>[14](https://encyclopediaofmath.org/wiki/Probability_theory)</sup>

**Caveats.** Although regular conditional distributions are well-defined and widely used, they can violate intuition drawn from the classical definition of conditional probability given an event of positive probability; the literature introduces the notion of a proper r.c.d. and gives necessary and sufficient conditions under which a given probability kernel is a proper r.c.d. with respect to a unique sub-\( \sigma \)-algebra.<sup>[15](https://arxiv.org/abs/2503.20489)</sup>

## References

1. [Conditional probability distribution, StatLect](https://www.statlect.com/fundamentals-of-probability/conditional-probability-distributions)
2. [Conditional density, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Conditional_density)
3. [Probability Theory 2 Lecture Notes, Bridgewater State University](https://webhost.bridgew.edu/j1pike/Prob%20Theory%20II%20notes.pdf)
4. [MAT 235B lecture notes, J. Gravner, UC Davis](https://www.math.ucdavis.edu/~gravner/MAT235B/assignments/MAT235B_lec1.pdf)
5. [Existence of Regular Conditional Distributions on Standard Borel Spaces, Theorem #17345, Androma](https://androma.org/theorems/17345)
6. [Conditional distributions (alternate StatLect mirror)](https://new.statlect.com/fundamentals-of-probability/conditional-probability-distributions)
7. [Mathlib.Probability.Kernel.CondDistrib documentation](https://math.iisc.ac.in/~gadgil/proofs-and-programs-2023/doc/Mathlib/Probability/Kernel/CondDistrib.html)
8. [Lecture 3: Conditional distributions, CMU 36-754, C. Shalizi](https://www.stat.cmu.edu/~cshalizi/754/notes/lecture-03.pdf)
9. [Chapter 9. Conditional Expectation, Probability Theory I notes, UT Austin](https://web.ma.utexas.edu/users/gordanz/notes/theory_of_probability_I.pdf)
10. [Markov kernel, nLab](https://ncatlab.org/nlab/show/Markov%2Bkernel)
11. [Stochastic point process, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Stochastic_point_process)
12. [Standard probability space, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Standard_probability_space)
13. [de Finetti's theorem and the existence of regular conditional distributions and strong laws on exchangeable algebras, arXiv:2312.16349](https://arxiv.org/abs/2312.16349)
14. [Probability theory, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Probability_theory)
15. [On some properties of probability kernels, arXiv:2503.20489](https://arxiv.org/abs/2503.20489)
16. [galton.uchicago.edu](https://galton.uchicago.edu/~lalley/Courses/383/ConditionalExpectation.pdf)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Probability theory › Conditional probability and independence › Conditional distributions (static)*

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