Math 2.0

Probability theory and stochastic processes Definition

Normal distribution

Builds on Probability density functionDirac measureDensity of the normal distribution

Definition

Let μ∈ℝ\mu \in \mathbb{R} and σ2>0\sigma^2 > 0. The normal distribution with mean μ\mu and variance σ2\sigma^2, written 𝒩(μ,σ2)\mathcal{N}(\mu, \sigma^2), is the distribution on ℝ\mathbb{R} with the probability density function φμ,σ2(x)=12πσ2exp⁡(−(x−μ)22σ2),x∈ℝ.\varphi_{\mu, \sigma^2}(x) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{(x - \mu)^2}{2\sigma^2}\right), \quad x \in \mathbb{R}. The standard normal distribution is 𝒩(0,1)\mathcal{N}(0, 1).

Remarks

The function φμ,σ2\varphi_{\mu, \sigma^2} is the density of the normal distribution. Some authors, and the library Mathlib, also allow σ2=0\sigma^2 = 0. Then 𝒩(μ,0)\mathcal{N}(\mu, 0) is the Dirac measure δμ\delta_\mu. If X∼𝒩(μ,σ2)X \sim \mathcal{N}(\mu, \sigma^2) and a≠0a \ne 0, bb are real, then aX+b∼𝒩(aμ+b,a2σ2)aX + b \sim \mathcal{N}(a\mu + b, a^2 \sigma^2). In particular, (X−μ)/σ∼𝒩(0,1)(X - \mu)/\sigma \sim \mathcal{N}(0, 1). The cumulative distribution function of 𝒩(0,1)\mathcal{N}(0, 1) is written Φ\Phi. It has no formula with elementary functions.

Reading path

To understand this page from the ground up, read these 46 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. Indicator function definition
  32. Preimage of a set definition
  33. Sigma-algebra definition
  34. Event definition
  35. Borel sigma-algebra definition
  36. Probability axioms concept
  37. Probability measure definition
  38. Probability distribution definition
  39. Dirac measure definition
  40. Probability space definition
  41. Random variable definition
  42. Cumulative distribution function definition
  43. Random variable with a density definition
  44. Probability density function definition
  45. Density of the normal distribution definition
  46. Normal distribution definition

Where this leads

44 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Normal_distribution, General normal distribution
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Kinddefinition
WritingW1 stated
MSC60E05
DefinesNormal distribution
UsesProbability density function, Dirac measure, Density of the normal distribution

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