Math 2.0

Probability theory and stochastic processes Proposition

Mean of the normal distribution

Builds on Normal distributionExpected valueDensity of the normal distribution

Statement

Let μ∈ℝ\mu \in \mathbb{R}, let σ2>0\sigma^2 > 0 and let XX have the normal distribution 𝒩(μ,σ2)\mathcal{N}(\mu, \sigma^2). Then XX is integrable and E[X]=∫−∞∞x 12πσ2e−(x−μ)2/(2σ2) dx=μ.E[X] = \int_{-\infty}^{\infty} x \, \frac{1}{\sqrt{2\pi\sigma^2}} e^{-(x - \mu)^2/(2\sigma^2)} \, dx = \mu.

Remarks

Proof: substitute x=μ+σzx = \mu + \sigma z with σ=σ2\sigma = \sqrt{\sigma^2}. The integral becomes μ∫φ(z) dz+σ∫zφ(z) dz\mu \int \varphi(z) \, dz + \sigma \int z \varphi(z) \, dz, where φ\varphi is the standard normal density. The first integral is 11. The second is 00, because zφ(z)z \varphi(z) is an odd integrable function. So the parameter μ\mu is the mean. It is also the median and the mode, because the density is symmetric about μ\mu and has its maximum there.

Reading path

To understand this page from the ground up, read these 51 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. Injective function definition
  33. Preimage of a set definition
  34. Sigma-algebra definition
  35. Countable set definition
  36. Event definition
  37. Borel sigma-algebra definition
  38. Probability axioms concept
  39. Probability measure definition
  40. Dirac measure definition
  41. Probability distribution definition
  42. Probability space definition
  43. Random variable definition
  44. Discrete random variable definition
  45. Cumulative distribution function definition
  46. Random variable with a density definition
  47. Expected value definition
  48. Probability density function definition
  49. Density of the normal distribution definition
  50. Normal distribution definition
  51. Mean 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, Moments
About this page
Tag09PR1V6RWR
URIm2://09PR1V6RWR
Kindstatement, proposition
WritingW0 stub
MSC60E05
StatusThis copy of the site has no build data. Run m2 status 09PR1V6RWR.
UsesNormal distribution, Expected value, Density of the normal distribution

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