Math 2.0

Probability theory and stochastic processes Definition

Cauchy distribution

Builds on Probability density function

Definition

Let x0∈ℝx_0 \in \mathbb{R} and γ>0\gamma > 0. The Cauchy distribution with location x0x_0 and scale γ\gamma is the distribution on ℝ\mathbb{R} with the probability density function f(x)=1πγ(1+(x−x0γ)2)=γπ((x−x0)2+γ2),x∈ℝ.f(x) = \frac{1}{\pi \gamma \left(1 + \left(\frac{x - x_0}{\gamma}\right)^2\right)} = \frac{\gamma}{\pi \left((x - x_0)^2 + \gamma^2\right)}, \quad x \in \mathbb{R}. The standard Cauchy distribution has x0=0x_0 = 0 and γ=1\gamma = 1, with density 1π(1+x2)\frac{1}{\pi(1 + x^2)}.

Remarks

The cumulative distribution function is F(x)=12+1πarctan⁡(x−x0γ)F(x) = \frac{1}{2} + \frac{1}{\pi} \arctan\left(\frac{x - x_0}{\gamma}\right). The median is x0x_0. The integral ∫∣x∣f(x) dx\int |x| f(x) \, dx is infinite, so the distribution has no mean. It also has no variance. If X1,…,XnX_1, \dots, X_n are independent standard Cauchy, then their average (X1+⋯+Xn)/n(X_1 + \dots + X_n)/n is again standard Cauchy. So the law of large numbers fails without a finite mean. The ratio X/YX/Y of two independent standard normal random variables is standard Cauchy.

Reading path

To understand this page from the ground up, read these 43 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. Cumulative distribution function definition
  41. Random variable with a density definition
  42. Probability density function definition
  43. Cauchy distribution definition

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cauchy_distribution, Probability density function (PDF)
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