definition 11.1 Discrete probability space

open in the book · parts/02-mathematical-methods/09-probability-statistics.tex:68 · p. 371

ground object -- no derivation owed

Rests on

depends_on definition 7.45 Series ¶
- depends_on definition 7.2 Absolute value ¶
- depends_on definition 7.4 Convergence ¶
  - depends_on definition 7.2 Absolute value ¶ ↺

Supports

depends_on definition 11.3 Discrete random variable ¶
- depends_on definition 11.4 Expectation ¶
  - depends_on definition A.741 The statistical setting ¶
  - depends_on definition 11.23 Expectation, moments and quantiles in the continuous case ¶
    - depends_on example 11.30 Exponential lifetimes ¶
    - depends_on proposition 11.29 Gaussian moments ¶
    - depends_on proposition 11.27 Probability integral transform ¶
    - depends_on proposition 11.24 The algebra of the discrete case transfers ¶
      - depends_on example 11.30 Exponential lifetimes ¶ ↺
      - depends_on proposition 11.26 Product rule ¶
  - depends_on proposition 11.5 Transfer, linearity, monotonicity ¶
    - depends_on definition 11.11 Covariance and correlation ¶
      - depends_on corollary 11.15 Bounds on the correlation coefficient ¶
      - depends_on definition 11.16 Sample mean, variance and correlation ¶
      - depends_on lemma 11.63 Cauchy–Schwarz for random variables ¶
      - depends_on proposition 11.12 Bilinearity ¶
    - depends_on definition 11.6 Moments, variance, standard deviation ¶
      - depends_on definition 11.11 Covariance and correlation ¶ ↺
      - depends_on definition 11.16 Sample mean, variance and correlation ¶ ↺
      - depends_on example 11.18 Three standard discrete distributions ¶
      - depends_on lemma 11.63 Cauchy–Schwarz for random variables ¶ ↺
      - depends_on lemma 11.50 Markov and Chebyshev inequalities ¶
      - depends_on proposition 11.7 Variance identity ¶
    - depends_on lemma 11.50 Markov and Chebyshev inequalities ¶ ↺
    - depends_on proposition 11.12 Bilinearity ¶ ↺
    - depends_on proposition 11.24 The algebra of the discrete case transfers ¶ ↺
    - depends_on proposition 11.7 Variance identity ¶ ↺
  - depends_on theorem A.169 Dominated convergence ¶
  - depends_on theorem A.170 Fubini's theorem, bounded case ¶
  - depends_on theorem A.223 Tonelli and dominated convergence, as used here ¶
- depends_on definition 11.13 Independence ¶
  - depends_on definition 11.25 Joint density and independence ¶
    - depends_on proposition 11.26 Product rule ¶ ↺
  - depends_on definition 11.16 Sample mean, variance and correlation ¶ ↺
  - depends_on proposition 11.14 Independence implies zero covariance; the converse fails ¶
    - depends_on example 11.18 Three standard discrete distributions ¶ ↺
    - depends_on proposition 11.17 Sampling identities ¶
      - depends_on example 11.19 A counting measurement ¶
      - depends_on example 11.31 A length measurement ¶
- depends_on definition 11.8 Quantiles and median ¶
  - depends_on example 11.31 A length measurement ¶ ↺
  - depends_on proposition 11.10 What the mean and the median minimise ¶
    - depends_on example 11.30 Exponential lifetimes ¶ ↺
depends_on definition 11.34 Measure; measure space ¶
- depends_on definition 11.38 Measure-preserving map; invariant measure; ergodicity ¶
  - depends_on theorem 11.39 Birkhoff's pointwise ergodic theorem ¶
- depends_on definition 11.35 Null set; almost everywhere ¶
  - depends_on definition 11.38 Measure-preserving map; invariant measure; ergodicity ¶ ↺
  - depends_on proposition 11.36 Monotonicity, countable subadditivity, and null sets ¶
    - depends_on theorem 11.39 Birkhoff's pointwise ergodic theorem ¶ ↺
- depends_on proposition 11.36 Monotonicity, countable subadditivity, and null sets ¶ ↺
depends_on definition 11.33 Sigma-algebra ¶
- depends_on definition 11.34 Measure; measure space ¶ ↺
depends_on proposition 11.2 Elementary rules ¶
- depends_on proposition 11.5 Transfer, linearity, monotonicity ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Series	declared	parts/02-mathematical-methods/09-probability-statistics.tex:79
`depends_on`	←	Discrete random variable	declared	parts/02-mathematical-methods/09-probability-statistics.tex:131
`depends_on`	←	Measure; measure space	declared	parts/02-mathematical-methods/09-probability-statistics.tex:898
`depends_on`	←	Sigma-algebra	declared	parts/02-mathematical-methods/09-probability-statistics.tex:880
`depends_on`	←	Elementary rules	declared	parts/02-mathematical-methods/09-probability-statistics.tex:99