Math 2.0

Probability theory and stochastic processes Theorem

Moments are derivatives of the moment-generating function at zero

Builds on Moment-generating functionMoment

Statement

Let XX be a real random variable, and let h>0h > 0 with MX(t)=E[etX]<∞M_X(t) = E[e^{tX}] < \infty for all t∈(−h,h)t \in (-h, h). Then XX has finite moments of all orders, the moment-generating function MXM_X is infinitely differentiable on (−h,h)(-h, h), and for each integer k≥0k \ge 0, MX(k)(0)=E[Xk].M_X^{(k)}(0) = E\left[X^k\right]. Moreover, MX(t)=∑k=0∞E[Xk] tk/k!M_X(t) = \sum_{k=0}^{\infty} E[X^k] \, t^k / k! for ∣t∣<h|t| < h.

Remarks

The proof differentiates under the expectation sign. The bound ∣x∣ketx≤C(eax+e−ax)|x|^k e^{t x} \le C(e^{a x} + e^{-a x}) for ∣t∣<a<h|t| < a < h justifies this by dominated convergence. Example: for X∼Exp⁡(λ)X \sim \operatorname{Exp}(\lambda), MX(t)=λ/(λ−t)M_X(t) = \lambda/(\lambda - t), so MX′(0)=1/λM_X'(0) = 1/\lambda and MX′′(0)=2/λ2M_X''(0) = 2/\lambda^2. So Var⁡(X)=1/λ2\operatorname{Var}(X) = 1/\lambda^2.

Reading path

To understand this page from the ground up, read these 52 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. Injective function definition
  32. Preimage of a set definition
  33. Sigma-algebra definition
  34. Countable set definition
  35. Event definition
  36. Borel sigma-algebra definition
  37. Probability axioms concept
  38. Probability measure definition
  39. Probability distribution definition
  40. Probability space definition
  41. Random variable definition
  42. Cumulative distribution function definition
  43. Discrete random variable definition
  44. Law of a random variable definition
  45. Random variable with a density definition
  46. Probability mass function definition
  47. Expected value definition
  48. Probability density function definition
  49. Law of the unconscious statistician statement
  50. Moment definition
  51. Moment-generating function definition
  52. Moments are derivatives of the moment-generating function at zero statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Moment_generating_function, Calculations of moments
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Kindstatement, theorem
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MSC60E10
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