Math 2.0

Probability theory and stochastic processes Definition

Chi-squared distribution

Builds on Normal distributionIndependent and identically distributed random variablesProbability density functionGamma distributionSample variance

Definition

Let k≥1k \ge 1 be an integer and let Z1,…,ZkZ_1, \dots, Z_k be i.i.d. standard normal random variables. The chi-squared distribution with kk degrees of freedom, written χk2\chi^2_k, is the law of Q=Z12+Z22+⋯+Zk2.Q = Z_1^2 + Z_2^2 + \dots + Z_k^2.

Remarks

It has the density f(x)=xk/2−1e−x/22k/2 Γ(k/2)f(x) = \frac{x^{k/2 - 1} e^{-x/2}}{2^{k/2}\, \Gamma(k/2)} for x>0x > 0, and f(x)=0f(x) = 0 for x≤0x \le 0. So χk2\chi^2_k is the gamma distribution with shape k/2k/2 and rate 1/21/2. If Q∼χk2Q \sim \chi^2_k, then E[Q]=kE[Q] = k and Var⁡(Q)=2k\operatorname{Var}(Q) = 2k. If Q1∼χk12Q_1 \sim \chi^2_{k_1} and Q2∼χk22Q_2 \sim \chi^2_{k_2} are independent, then Q1+Q2∼χk1+k22Q_1 + Q_2 \sim \chi^2_{k_1 + k_2}. For an i.i.d. 𝒩(μ,σ2)\mathcal{N}(\mu, \sigma^2) sample of size nn with sample variance Sn2S_n^2, (n−1)Sn2/σ2∼χn−12(n - 1) S_n^2 / \sigma^2 \sim \chi^2_{n - 1}.

Reading path

To understand this page from the ground up, read these 77 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. Conditional probability definition
  44. Random variable definition
  45. Discrete random variable definition
  46. Cumulative distribution function definition
  47. Independence of two events definition
  48. Law of a random variable definition
  49. Random variable with a density definition
  50. Joint probability distribution definition
  51. Independence of two random variables definition
  52. Mutual independence of events definition
  53. Expected value definition
  54. Probability density function definition
  55. Identically distributed random variables definition
  56. Marginal distribution definition
  57. Variance definition
  58. Independent sigma-algebras definition
  59. Linearity of expectation statement
  60. Density of the normal distribution definition
  61. Exponential distribution definition
  62. Independence means the joint law is the product of the marginals statement
  63. Independence of random variables definition
  64. Covariance definition
  65. Normal distribution definition
  66. Gamma distribution definition
  67. Expectation of a product of independent random variables statement
  68. Variance of a sum statement
  69. Independent and identically distributed random variables definition
  70. Independent random variables are uncorrelated statement
  71. Statistic definition
  72. Variance of a sum of independent random variables statement
  73. Estimator definition
  74. Sample mean definition
  75. Bias of an estimator definition
  76. Sample variance definition
  77. Chi-squared distribution definition

Where this leads

14 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Chi-squared_distribution, Definitions
  2. Web https://en.wikipedia.org/wiki/Chi-squared_distribution, Probability density function
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MSC60E05, 62E15
DefinesChi-squared distribution
UsesNormal distribution, Independent and identically distributed random variables, Probability density function, Gamma distribution, Sample variance

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