Math 2.0

Probability theory and stochastic processes Definition

Moment-generating function

Builds on Expected valueMoment

Definition

Let XX be a real random variable. The moment-generating function of XX is the function MX(t)=E[etX],t∈ℝ,M_X(t) = E\left[e^{tX}\right], \quad t \in \mathbb{R}, with values in (0,∞](0, \infty].

Remarks

Always MX(0)=1M_X(0) = 1. The set of tt with MX(t)<∞M_X(t) < \infty is an interval that contains 00. If MXM_X is finite on an open interval around 00, then all moments of XX are finite and MXM_X determines the law of XX. Some random variables have MX(t)=∞M_X(t) = \infty for all t≠0t \ne 0. An example is a Cauchy random variable. For X∼Exp⁡(λ)X \sim \operatorname{Exp}(\lambda), MX(t)=λ/(λ−t)M_X(t) = \lambda / (\lambda - t) for t<λt < \lambda, and MX(t)=∞M_X(t) = \infty for t≥λt \ge \lambda.

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. 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

Where this leads

11 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Moment_generating_function, Definition
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DefinesMoment-generating function
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