Math 2.0

Statistics Definition

Joint probability distribution

Builds on Random variableLaw of a random variableCartesian product

Definition

Let X1,…,XnX_1, \dots, X_n be real random variables on one probability space (Ω,ℱ,P)(\Omega, \mathcal{F}, P). Their joint distribution is the law of the random vector (X1,…,Xn) ⁣:Ω→ℝn(X_1, \dots, X_n) \colon \Omega \to \mathbb{R}^n. It is the probability measure on the Borel sets of ℝn\mathbb{R}^n with μ(B)=P((X1,…,Xn)∈B).\mu(B) = P((X_1, \dots, X_n) \in B). Their joint cumulative distribution function is F(x1,…,xn)=P(X1≤x1,…,Xn≤xn)F(x_1, \dots, x_n) = P(X_1 \le x_1, \dots, X_n \le x_n).

Remarks

For discrete variables, the joint probability mass function p(x1,…,xn)=P(X1=x1,…,Xn=xn)p(x_1, \dots, x_n) = P(X_1 = x_1, \dots, X_n = x_n) determines the joint distribution. The variables have a joint density ff if μ(B)=∫Bf(x) dx\mu(B) = \int_B f(x) \, dx for each Borel set B⊆ℝnB \subseteq \mathbb{R}^n. The joint distribution determines the marginal distributions, but the marginal distributions do not determine the joint distribution. For two rolls of a fair die, the joint mass function of the two results is 1/361/36 at each pair (i,j)(i, j) with 1≤i,j≤61 \le i, j \le 6.

Reading path

To understand this page from the ground up, read these 41 pages in order. The foundations come first.

  1. Construction of the reals from Cauchy sequences concept
  2. Element definition
  3. Integers object
  4. Ordered field definition
  5. Proposition concept
  6. Random experiment concept
  7. Rational numbers object
  8. Real numbers object
  9. Set definition
  10. Ring of integers statement
  11. Disjunction definition
  12. Implication definition
  13. Predicate definition
  14. Ordered pair definition
  15. Union definition
  16. Sample space definition
  17. The rationals form a field statement
  18. Existential quantifier definition
  19. Universal quantifier definition
  20. Cartesian product definition
  21. Subset definition
  22. The rationals form an ordered field statement
  23. Uniqueness quantifier definition
  24. Binary relation definition
  25. Universal set definition
  26. Order on the reals definition
  27. Power set definition
  28. Complement definition
  29. Function definition
  30. Interval definition
  31. Preimage of a set definition
  32. Sigma-algebra definition
  33. Event definition
  34. Borel sigma-algebra definition
  35. Probability axioms concept
  36. Probability measure definition
  37. Probability distribution definition
  38. Probability space definition
  39. Random variable definition
  40. Law of a random variable definition
  41. Joint probability distribution definition

Where this leads

69 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Joint_probability_distribution, Joint cumulative distribution function
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