definition 11.4 Expectation

open in the book · parts/02-mathematical-methods/09-probability-statistics.tex:134 · p. 372

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 ¶ ↺
depends_on definition 11.3 Discrete random variable ¶
- depends_on definition 11.1 Discrete probability space ¶
  - depends_on definition 7.45 Series ¶ ↺

Supports

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 example 11.19 A counting measurement ¶
    - depends_on example 11.31 A length measurement ¶
    - 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 lemma 11.63 Cauchy–Schwarz for random variables ¶
    - depends_on corollary 11.15 Bounds on the correlation coefficient ¶ ↺
    - depends_on corollary 11.65 When the bound is attained ¶
    - depends_on lemma A.199 The standardised third moment is at least one ¶
      - depends_on lemma A.200 Transform estimate ¶
    - depends_on theorem 11.64 Cramér–Rao inequality ¶
      - depends_on corollary 11.65 When the bound is attained ¶ ↺
      - depends_on example 11.62 Information in a counting experiment ¶
  - depends_on proposition 11.12 Bilinearity ¶
    - depends_on example 11.18 Three standard discrete distributions ¶
      - depends_on example 11.19 A counting measurement ¶ ↺
    - 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 proposition 11.17 Sampling identities ¶ ↺
- 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 corollary 11.51 Weak law of large numbers ¶
      - depends_on theorem A.205 Weak law under integrability alone ¶
      - depends_on theorem 11.67 Asymptotics of the MLE ¶
    - depends_on theorem 11.101 Davies' bound ¶
      - depends_on corollary 11.105 Trials factor at high significance ¶
      - depends_on example 11.107 Degrading a local five sigma ¶
  - depends_on proposition 11.7 Variance identity ¶
    - depends_on proposition 11.10 What the mean and the median minimise ¶
      - depends_on example 11.30 Exponential lifetimes ¶ ↺
- 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 ¶

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:142
`depends_on`	→	Discrete random variable	declared	parts/02-mathematical-methods/09-probability-statistics.tex:142
`depends_on`	←	The statistical setting	declared	appendices/A-long-proofs.tex:35945
`depends_on`	←	Expectation, moments and quantiles in the continuous case	declared	parts/02-mathematical-methods/09-probability-statistics.tex:631
`depends_on`	←	Transfer, linearity, monotonicity	declared	parts/02-mathematical-methods/09-probability-statistics.tex:159
`depends_on`	←	Dominated convergence	declared	appendices/A-long-proofs.tex:8907
`depends_on`	←	Fubini's theorem, bounded case	declared	appendices/A-long-proofs.tex:8918
`depends_on`	←	Tonelli and dominated convergence, as used here	declared	appendices/A-long-proofs.tex:11302