Math 2.0

Probability theory and stochastic processes Definition

Student's t-distribution

Builds on Normal distributionChi-squared distributionIndependence of random variablesCauchy distributionSample meanSample variance

Definition

Let ν>0\nu > 0 be a real number. Student's t-distribution with ν\nu degrees of freedom, written tνt_\nu, is the law on ℝ\mathbb{R} with the density f(t)=Γ(ν+12)νπ Γ(ν2)(1+t2ν)−(ν+1)/2,t∈ℝ.f(t) = \frac{\Gamma\left(\frac{\nu + 1}{2}\right)}{\sqrt{\nu \pi}\, \Gamma\left(\frac{\nu}{2}\right)} \left(1 + \frac{t^2}{\nu}\right)^{-(\nu + 1)/2}, \qquad t \in \mathbb{R}.

Remarks

Let ZZ be standard normal and V∼χν2V \sim \chi^2_\nu (see chi-squared distribution), independent. Then Z/V/ν∼tνZ / \sqrt{V / \nu} \sim t_\nu. For an i.i.d. 𝒩(μ,σ2)\mathcal{N}(\mu, \sigma^2) sample of size n≥2n \ge 2, Xˉn−μSn/n∼tn−1\frac{\bar{X}_n - \mu}{S_n / \sqrt{n}} \sim t_{n - 1}, with the sample mean Xˉn\bar{X}_n and the sample standard deviation SnS_n. The mean is 00 for ν>1\nu > 1, and the variance is ν/(ν−2)\nu / (\nu - 2) for ν>2\nu > 2. For ν=1\nu = 1, t1t_1 is the standard Cauchy distribution. As ν→∞\nu \to \infty, the density of tνt_\nu converges to the standard normal density.

Reading path

To understand this page from the ground up, read these 79 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. Cauchy distribution definition
  61. Density of the normal distribution definition
  62. Exponential distribution definition
  63. Independence means the joint law is the product of the marginals statement
  64. Independence of random variables definition
  65. Covariance definition
  66. Normal distribution definition
  67. Gamma distribution definition
  68. Expectation of a product of independent random variables statement
  69. Variance of a sum statement
  70. Independent and identically distributed random variables definition
  71. Independent random variables are uncorrelated statement
  72. Statistic definition
  73. Variance of a sum of independent random variables statement
  74. Estimator definition
  75. Sample mean definition
  76. Bias of an estimator definition
  77. Sample variance definition
  78. Chi-squared distribution definition
  79. Student's t-distribution definition

Where this leads

4 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Student's_t-distribution, Probability density function
  2. Web https://en.wikipedia.org/wiki/Student's_t-distribution, Sampling distribution of t-statistic
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Kinddefinition
WritingW1 stated
MSC60E05, 62E15
DefinesStudent's t-distribution
UsesNormal distribution, Chi-squared distribution, Independence of random variables, Cauchy distribution, Sample mean, Sample variance

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