Math 2.0

Probability theory and stochastic processes Proposition

Moment-generating function of a sum of independent random variables

Builds on Moment-generating functionExpectation of a product of independent random variablesIndependence of random variables

Statement

Let X1,…,XnX_1, \dots, X_n be independent real random variables, and let S=X1+⋯+XnS = X_1 + \dots + X_n. Then for each t∈ℝt \in \mathbb{R}, the moment-generating functions satisfy MS(t)=MX1(t) MX2(t)⋯MXn(t),M_S(t) = M_{X_1}(t) \, M_{X_2}(t) \cdots M_{X_n}(t), in (0,∞](0, \infty].

Remarks

Proof: etS=etX1⋯etXne^{tS} = e^{tX_1} \cdots e^{tX_n}. The factors are nonnegative and independent, so the product rule for expectations applies. For i.i.d. XiX_i, MS(t)=MX1(t)nM_S(t) = M_{X_1}(t)^n. Example: if Xi∼Exp⁡(λ)X_i \sim \operatorname{Exp}(\lambda) are independent, then MS(t)=(λ/(λ−t))nM_S(t) = (\lambda/(\lambda - t))^n for t<λt < \lambda. This is the moment-generating function of the gamma distribution Gamma⁡(n,λ)\operatorname{Gamma}(n, \lambda).

Reading path

To understand this page from the ground up, read these 62 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. Conditional probability definition
  42. Random variable definition
  43. Cumulative distribution function definition
  44. Discrete random variable definition
  45. Law of a random variable definition
  46. Independence of two events definition
  47. Random variable with a density definition
  48. Joint probability distribution definition
  49. Probability mass function definition
  50. Mutual independence of events definition
  51. Expected value definition
  52. Independence of two random variables definition
  53. Probability density function definition
  54. Marginal distribution definition
  55. Independent sigma-algebras definition
  56. Law of the unconscious statistician statement
  57. Independence of random variables definition
  58. Independence means the joint law is the product of the marginals statement
  59. Moment definition
  60. Moment-generating function definition
  61. Expectation of a product of independent random variables statement
  62. Moment-generating function of a sum of independent random variables statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Moment_generating_function, Linear combination of independent random variables
About this page
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