definition 11.20 Density

open in the book · parts/02-mathematical-methods/09-probability-statistics.tex:572 · p. 377

ground object -- no derivation owed

Rests on

depends_on definition 7.39 Darboux sums and the definite integral ¶
- depends_on axiom 7.1 Completeness of $\R$ ¶

Supports

depends_on definition 11.21 Distribution function ¶
- depends_on definition A.174 Tightness ¶
  - depends_on lemma A.175 Tightness makes the Helly limit proper ¶
  - depends_on lemma A.180 Tightness ¶
- depends_on lemma 11.52 Slutsky ¶
  - depends_on theorem A.206 Cramér–Wold, and continuous mapping ¶
    - 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 A.214 Wilks, $k$ parameters and $r$ constraints ¶
  - 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 11.88 Wilks ¶ ↺
- depends_on proposition 11.22 Properties of the distribution function ¶
  - depends_on proposition 11.27 Probability integral transform ¶
- depends_on theorem A.168 Lévy's continuity theorem ¶
  - depends_on corollary A.181 The converse ¶
- depends_on theorem A.173 Helly's selection theorem ¶
  - depends_on lemma A.176 Helly–Bray ¶
    - depends_on corollary A.181 The converse ¶ ↺
  - depends_on lemma A.175 Tightness makes the Helly limit proper ¶ ↺
- depends_on theorem A.206 Cramér–Wold, and continuous mapping ¶ ↺
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 definition 11.28 Gaussian distribution ¶
- depends_on definition A.218 The hypotheses ¶
  - depends_on lemma A.220 At each point, field and derivative are independent Gaussians ¶
  - depends_on theorem A.219 Rice's formula ¶
    - depends_on corollary A.227 The stationary case ¶
- depends_on lemma A.226 The level factor ¶
- depends_on lemma A.225 The mean positive part ¶
- depends_on proposition 11.29 Gaussian moments ¶ ↺
- depends_on theorem A.192 Berry–Esseen ¶
- depends_on theorem A.183 Lindeberg–Feller ¶
  - depends_on corollary A.188 Lindeberg–Lévy is the special case ¶
  - depends_on corollary A.189 Lyapunov's condition ¶
- depends_on theorem 11.102 Rice's formula ¶
  - depends_on theorem 11.103 Level dependence of the mean upcrossing count ¶
    - depends_on corollary 11.105 Trials factor at high significance ¶
    - depends_on example 11.107 Degrading a local five sigma ¶
depends_on definition 11.25 Joint density and independence ¶
- depends_on proposition 11.26 Product rule ¶ ↺

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`	→	Darboux sums and the definite integral	declared	parts/02-mathematical-methods/09-probability-statistics.tex:582
`depends_on`	←	Distribution function	declared	parts/02-mathematical-methods/09-probability-statistics.tex:592
`depends_on`	←	Expectation, moments and quantiles in the continuous case	declared	parts/02-mathematical-methods/09-probability-statistics.tex:631
`depends_on`	←	Gaussian distribution	declared	parts/02-mathematical-methods/09-probability-statistics.tex:732
`depends_on`	←	Joint density and independence	declared	parts/02-mathematical-methods/09-probability-statistics.tex:674