Math 2.0

Probability theory and stochastic processes Proposition

Moment-generating function of the normal distribution

Builds on Moment-generating functionNormal distributionDensity 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}, MX(t)=E[etX]=exp⁡(μt+12σ2t2).M_X(t) = E\left[e^{tX}\right] = \exp\left(\mu t + \tfrac{1}{2} \sigma^2 t^2\right).

Remarks

Proof for Z∼𝒩(0,1)Z \sim \mathcal{N}(0, 1): complete the square, tz−z2/2=t2/2−(z−t)2/2tz - z^2/2 = t^2/2 - (z - t)^2/2. So E[etZ]=et2/2∫φ(z−t) dz=et2/2E[e^{tZ}] = e^{t^2/2} \int \varphi(z - t) \, dz = e^{t^2/2}. The general case follows from X=μ+σZX = \mu + \sigma Z with σ=σ2\sigma = \sqrt{\sigma^2}: MX(t)=eμtMZ(σt)M_X(t) = e^{\mu t} M_Z(\sigma t). The moment-generating function is finite for all tt. Its derivatives at 00 give E[X]=μE[X] = \mu and E[X2]=μ2+σ2E[X^2] = \mu^2 + \sigma^2. The logarithm μt+σ2t2/2\mu t + \sigma^2 t^2 / 2 is a polynomial of degree 22. So all cumulants of order 33 or more are 00.

Reading path

To understand this page from the ground up, read these 56 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. 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. Density of the normal distribution definition
  53. Moment definition
  54. Normal distribution definition
  55. Moment-generating function definition
  56. Moment-generating function of the normal distribution statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Normal_distribution, Moment- and cumulant-generating functions
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Tag0B1G4E21G7
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Kindstatement, proposition
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MSC60E10
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UsesMoment-generating function, Normal distribution, Density of the normal distribution

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