Math 2.0

Probability theory and stochastic processes Definition

Characteristic function

Builds on Expected valueLaw of the unconscious statisticianEuler's formulaComplex numbers

Definition

Let XX be a real random variable with law μX\mu_X. Let ii be the imaginary unit in the complex numbers. The characteristic function of XX is the function φX ⁣:ℝ→ℂ\varphi_X \colon \mathbb{R} \to \mathbb{C} with φX(t)=E[eitX]=E[cos⁡(tX)]+i E[sin⁡(tX)]=∫ℝeitx μX(dx).\varphi_X(t) = E\left[e^{itX}\right] = E[\cos(tX)] + i \, E[\sin(tX)] = \int_{\mathbb{R}} e^{itx} \, \mu_X(dx).

Remarks

The expectation exists for all tt, because ∣eitX∣=1|e^{itX}| = 1. The second form uses Euler's formula. Always φX(0)=1\varphi_X(0) = 1, ∣φX(t)∣≤1|\varphi_X(t)| \le 1, and φX\varphi_X is uniformly continuous on ℝ\mathbb{R}. For a,b∈ℝa, b \in \mathbb{R}, φaX+b(t)=eitbφX(at)\varphi_{aX + b}(t) = e^{itb} \varphi_X(at). If E[∣X∣k]<∞E[|X|^k] < \infty, then φX\varphi_X is kk times continuously differentiable and φX(k)(0)=ikE[Xk]\varphi_X^{(k)}(0) = i^k E[X^k].

Reading path

To understand this page from the ground up, read these 52 pages in order. The foundations come first.

  1. Complex numbers object
  2. Construction of the reals from Cauchy sequences concept
  3. Element definition
  4. Integers object
  5. Ordered field definition
  6. Proposition concept
  7. Random experiment concept
  8. Rational numbers object
  9. Real numbers object
  10. Euler's formula statement
  11. Set definition
  12. Ring of integers statement
  13. Disjunction definition
  14. Implication definition
  15. Predicate definition
  16. Ordered pair definition
  17. Union definition
  18. Sample space definition
  19. The rationals form a field statement
  20. Existential quantifier definition
  21. Universal quantifier definition
  22. Cartesian product definition
  23. Subset definition
  24. The rationals form an ordered field statement
  25. Uniqueness quantifier definition
  26. Binary relation definition
  27. Power set definition
  28. Universal set definition
  29. Order on the reals definition
  30. Function definition
  31. Complement definition
  32. Interval definition
  33. Injective function definition
  34. Preimage of a set definition
  35. Sigma-algebra definition
  36. Countable set definition
  37. Event definition
  38. Borel sigma-algebra definition
  39. Probability axioms concept
  40. Probability measure definition
  41. Probability distribution definition
  42. Probability space definition
  43. Random variable definition
  44. Cumulative distribution function definition
  45. Discrete random variable definition
  46. Law of a random variable definition
  47. Random variable with a density definition
  48. Probability mass function definition
  49. Expected value definition
  50. Probability density function definition
  51. Law of the unconscious statistician statement
  52. Characteristic function definition

Where this leads

15 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Characteristic_function_(probability_theory), Definition
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DefinesCharacteristic function
UsesExpected value, Law of the unconscious statistician, Euler's formula, Complex numbers

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