Math 2.0

Probability theory and stochastic processes Definition

Probability-generating function

Builds on Probability mass functionExpected valueGenerating function

Definition

Let XX be a random variable with values in {0,1,2,… }\{0, 1, 2, \dots\} and with mass function p(k)=P(X=k)p(k) = P(X = k). The probability-generating function of XX is GX(z)=E[zX]=∑k=0∞p(k) zk,G_X(z) = E\left[z^X\right] = \sum_{k=0}^{\infty} p(k) \, z^k, for each complex zz with ∣z∣≤1|z| \le 1.

Remarks

GXG_X is the generating function of the sequence p(0),p(1),…p(0), p(1), \dots. The series converges absolutely for ∣z∣≤1|z| \le 1, and GX(1)=1G_X(1) = 1. GXG_X determines the law of XX: P(X=k)=GX(k)(0)/k!P(X = k) = G_X^{(k)}(0)/k!. If E[X]<∞E[X] < \infty, then E[X]=GX′(1)E[X] = G_X'(1), with the left derivative at 11. Also E[X(X−1)]=GX′′(1)E[X(X - 1)] = G_X''(1) when it is finite. For X∼Poisson⁡(λ)X \sim \operatorname{Poisson}(\lambda), GX(z)=eλ(z−1)G_X(z) = e^{\lambda(z - 1)}. For X∼Bin⁡(n,p)X \sim \operatorname{Bin}(n, p), GX(z)=(1−p+pz)nG_X(z) = (1 - p + pz)^n.

Reading path

To understand this page from the ground up, read these 47 pages in order. The foundations come first.

  1. Construction of the reals from Cauchy sequences concept
  2. Element definition
  3. Formal power series definition
  4. Integers object
  5. Ordered field definition
  6. Proposition concept
  7. Random experiment concept
  8. Rational numbers object
  9. Real numbers object
  10. Set definition
  11. Generating function definition
  12. Ring of integers statement
  13. Disjunction definition
  14. Implication definition
  15. Predicate definition
  16. Ordered pair definition
  17. Union definition
  18. Sample space definition
  19. The rationals form a field statement
  20. Existential quantifier definition
  21. Universal quantifier definition
  22. Cartesian product definition
  23. Subset definition
  24. The rationals form an ordered field statement
  25. Uniqueness quantifier definition
  26. Binary relation definition
  27. Universal set definition
  28. Order on the reals definition
  29. Power set definition
  30. Complement definition
  31. Function definition
  32. Interval definition
  33. Injective function definition
  34. Preimage of a set definition
  35. Sigma-algebra definition
  36. Countable set definition
  37. Event definition
  38. Borel sigma-algebra definition
  39. Probability axioms concept
  40. Probability measure definition
  41. Probability space definition
  42. Random variable definition
  43. Discrete random variable definition
  44. Random variable with a density definition
  45. Probability mass function definition
  46. Expected value definition
  47. Probability-generating function definition

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Probability_generating_function, Univariate case
About this page
Tag0Z8PSSAVBA
URIm2://0Z8PSSAVBA
Kinddefinition
WritingW0 stub
MSC60E05
DefinesProbability-generating function
UsesProbability mass function, Expected value, Generating function

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by