lemma 11.63 Cauchy–Schwarz for random variables

open in the book · parts/02-mathematical-methods/09-probability-statistics.tex:1797 · p. 392

Rests on

depends_on definition 11.11 Covariance and correlation ¶
- depends_on definition 11.6 Moments, variance, standard deviation ¶
  - depends_on proposition 11.5 Transfer, linearity, monotonicity ¶
    - depends_on definition 11.4 Expectation ¶
      - depends_on definition 7.45 Series ¶
      - depends_on definition 11.3 Discrete random variable ¶
    - 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 ¶
- depends_on proposition 11.5 Transfer, linearity, monotonicity ¶ ↺
depends_on definition 11.6 Moments, variance, standard deviation ¶ ↺
proves proof ch:09-probability-statistics@proof-24 ¶

Supports

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 ¶

Neighborhood

Every logical edge within two steps of this node.

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partly declared
a check failed
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Edges

type	direction	node	provenance	where
`depends_on`	→	Covariance and correlation	declared	parts/02-mathematical-methods/09-probability-statistics.tex:1806
`depends_on`	→	Moments, variance, standard deviation	declared	parts/02-mathematical-methods/09-probability-statistics.tex:1806
`depends_on`	←	Bounds on the correlation coefficient	declared	parts/02-mathematical-methods/09-probability-statistics.tex:407
`depends_on`	←	When the bound is attained	declared	parts/02-mathematical-methods/09-probability-statistics.tex:1892
`depends_on`	←	The standardised third moment is at least one	declared	appendices/A-long-proofs.tex:10139
`depends_on`	←	Cramér–Rao inequality	declared	parts/02-mathematical-methods/09-probability-statistics.tex:1849
`proves`	←	ch:09-probability-statistics@proof-24	declared	parts/02-mathematical-methods/09-probability-statistics.tex:1809