Math 2.0

Probability theory and stochastic processes Proposition

Characteristic function of the normal distribution

Builds on Characteristic functionNormal distributionMoment-generating function of the normal distribution

Statement

Let μ∈ℝ\mu \in \mathbb{R}, let σ2>0\sigma^2 > 0 and let X∼𝒩(μ,σ2)X \sim \mathcal{N}(\mu, \sigma^2). Then for each t∈ℝt \in \mathbb{R}, φX(t)=E[eitX]=exp⁡(iμt−12σ2t2).\varphi_X(t) = E\left[e^{itX}\right] = \exp\left(i \mu t - \tfrac{1}{2} \sigma^2 t^2\right). In particular, the standard normal distribution has φ(t)=e−t2/2\varphi(t) = e^{-t^2/2}.

Remarks

One proof: the moment-generating function E[ezX]=exp⁡(μz+σ2z2/2)E[e^{zX}] = \exp(\mu z + \sigma^2 z^2/2) holds for real zz. Both sides are entire functions of zz, so the identity extends to all complex zz, and one puts z=itz = it. Another proof: φ(t)=e−t2/2\varphi(t) = e^{-t^2/2} for 𝒩(0,1)\mathcal{N}(0, 1) is the solution of φ′(t)=−tφ(t)\varphi'(t) = -t \varphi(t) with φ(0)=1\varphi(0) = 1. Integration by parts gives this differential equation. So the standard normal density is, up to the factor 2π\sqrt{2\pi}, its own Fourier transform.

Reading path

To understand this page from the ground up, read these 60 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. Indicator function definition
  35. Preimage of a set definition
  36. Sigma-algebra definition
  37. Countable set definition
  38. Event definition
  39. Borel sigma-algebra definition
  40. Probability axioms concept
  41. Probability measure definition
  42. Probability distribution definition
  43. Dirac measure definition
  44. Probability space definition
  45. Random variable definition
  46. Cumulative distribution function definition
  47. Discrete random variable definition
  48. Law of a random variable definition
  49. Random variable with a density definition
  50. Probability mass function definition
  51. Expected value definition
  52. Probability density function definition
  53. Law of the unconscious statistician statement
  54. Density of the normal distribution definition
  55. Characteristic function definition
  56. Moment definition
  57. Normal distribution definition
  58. Moment-generating function definition
  59. Moment-generating function of the normal distribution statement
  60. Characteristic function of the normal distribution statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Normal_distribution, Fourier transform and characteristic function
About this page
Tag0NS0TDA4E2
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Kindstatement, proposition
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MSC60E10
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UsesCharacteristic function, Normal distribution, Moment-generating function of the normal distribution

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