definition 11.6 Moments, variance, standard deviation

open in the book · parts/02-mathematical-methods/09-probability-statistics.tex:182 · p. 373

ground object -- no derivation owed

Rests on

depends_on proposition 11.5 Transfer, linearity, monotonicity ¶
- depends_on definition 11.4 Expectation ¶
  - 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 ¶ ↺
- depends_on proposition 11.2 Elementary rules ¶
  - depends_on definition 11.1 Discrete probability space ¶ ↺
  - proves proof ch:09-probability-statistics@proof-1 ¶
- proves proof ch:09-probability-statistics@proof-2 ¶

Supports

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 11.16 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 lemma A.208 The three averages ¶
      - depends_on corollary A.210 Uniform quadratic approximation on the $n^{-1/2}$ scale ¶
      - depends_on lemma A.211 Root-$n$ localisation ¶
  - depends_on theorem 11.67 Asymptotics of the MLE ¶
    - depends_on proposition 11.72 Nuisance parameters cost information ¶
    - depends_on theorem 11.88 Wilks ¶
      - depends_on corollary 11.89 Degrees of freedom with nuisance parameters ¶
- 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 ¶

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`	→	Transfer, linearity, monotonicity	declared	parts/02-mathematical-methods/09-probability-statistics.tex:194
`depends_on`	←	Covariance and correlation	declared	parts/02-mathematical-methods/09-probability-statistics.tex:325
`depends_on`	←	Sample mean, variance and correlation	declared	parts/02-mathematical-methods/09-probability-statistics.tex:453
`depends_on`	←	Three standard discrete distributions	declared	parts/02-mathematical-methods/09-probability-statistics.tex:523
`depends_on`	←	Cauchy–Schwarz for random variables	declared	parts/02-mathematical-methods/09-probability-statistics.tex:1806
`depends_on`	←	Markov and Chebyshev inequalities	declared	parts/02-mathematical-methods/09-probability-statistics.tex:1408
`depends_on`	←	Variance identity	declared	parts/02-mathematical-methods/09-probability-statistics.tex:215