Marginal distribution

Builds on Joint probability distribution Law of a random variable

Definition

Let $\mu$ be a probability distribution on $\mathbb{R}^2$ , for example the joint distribution of real random variables $X$ and $Y$ . The first marginal distribution of $\mu$ is the distribution $\mu_1$ on $\mathbb{R}$ with $\mu_1(A) = \mu(A \times \mathbb{R})$ for each Borel set $A \subseteq \mathbb{R}$ . The second marginal is $\mu_2(B) = \mu(\mathbb{R} \times B)$ . If $\mu$ is the law of $(X, Y)$ , then $\mu_1$ is the law of $X$ and $\mu_2$ is the law of $Y$ .

Remarks

The same definition works on a product $E_1 \times E_2$ of measurable spaces, and for $n$ variables with the projections onto some of the coordinates. In the discrete case, $P(X = x) = \sum_{y} P(X = x, Y = y)$ . In the case of a joint density $f(x, y)$ , $X$ has the density $f_X(x) = \int_{-\infty}^{\infty} f(x, y) \, dy$ . The name comes from tables: the sums of the rows and the columns of a joint table are written in its margins.

Reading path

To understand this page from the ground up, read these 42 pages in order. The foundations come first.

Construction of the reals from Cauchy sequences concept
Element definition
Integers object
Ordered field definition
Proposition concept
Random experiment concept
Rational numbers object
Real numbers object
Set definition
Ring of integers statement
Disjunction definition
Implication definition
Predicate definition
Ordered pair definition
Union definition
Sample space definition
The rationals form a field statement
Existential quantifier definition
Universal quantifier definition
Cartesian product definition
Subset definition
The rationals form an ordered field statement
Uniqueness quantifier definition
Binary relation definition
Universal set definition
Order on the reals definition
Power set definition
Complement definition
Function definition
Interval definition
Preimage of a set definition
Sigma-algebra definition
Event definition
Borel sigma-algebra definition
Probability axioms concept
Probability measure definition
Probability distribution definition
Probability space definition
Random variable definition
Law of a random variable definition
Joint probability distribution definition
Marginal distribution definition

Where this leads

68 pages build on this one.

Tag	`0AXP2Z5N9H`
URI	`m2://0AXP2Z5N9H`
Kind	definition
Writing	W0 stub
MSC	62H05
Defines	Marginal distribution
Uses	Joint probability distribution, Law of a random variable

Marginal distribution

Definition

Remarks

Reading path

Where this leads

Further reading

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