Index: The Book of Statistical Proofs ▷ General Theorems

Proofs and Definitions on General Theorems in the StatProofBook, as of 2023-12-23.

Concept Definition Properties Relationships Other
Probability theory
Probability prob
prob-joint
prob-marg
prob-cond
prob-ind
prob-exc

Probability axioms prob-ax prob-mon
prob-emp
prob-comp
prob-exh
prob-exh2
prob-tot
Probability mass function pmf pmf-sumind
pmf-sifct
pmf-sdfct
pmf-invfct

Probability density function pdf pdf-sumind
pdf-sifct
pdf-sdfct
pdf-invfct
pdf-linfct
pdf-cdf
Cumulative distribution function cdf
cdf-joint
cdf-sumind
cdf-sifct
cdf-sdfct
cdf-pmf
cdf-pdf
cdf-pit
cdf-itm
cdf-dt
Quantile function qf   qf-cdf
Characteristic function cf cf-fct
Moment-generating function mgf mgf-fct   mgf-ltt
mgf-lincomb
Cumulant-generating function cgf
Probability-generating function pgf   pgf-mean pgf-zero
pgf-one
Expected value mean
mean-samp
mean-rvec
mean-rmat
mean-nnrvar
mean-nonneg
mean-lin
mean-mono
mean-mult
mean-tr
mean-qf
mean-prodsqr
mean-tot
mean-lotus
Variance var
var-samp
var-nonneg
var-const
var-inv
var-scal
var-sum
var-lincomb
var-mean var-tot
Skewness skew
skew-samp
skew-mean
Covariance cov
cov-samp
covmat
covmat-samp
covmat-cross
cov-ind
cov-symm
covmat-inv
covmat-scal
covmat-sum
covmat-symm
covmat-psd
cov-mean
cov-var
cov-corr
covmat-mean
covmat-corrmat
cov-tot
Correlation corr
corr-samp
corrmat
corrmat-samp
corr-range corr-z
Median med
Mode mode
Standard deviation std
Full width at half maximum fwhm
Minimum min
Maximum max
Moments mom
mom-raw
mom-cent
mom-stand
momraw-1st
momraw-2nd
momcent-1st
momcent-2nd
mom-mgf
Information theory
Shannon entropy ent
ent-cond
ent-joint
ent-cross
ent-nonneg
ent-conc
entcross-conv
gibbs-ineq
logsum-ineq
Differential entropy dent
dent-cond
dent-joint
dent-cross
dent-neg
dent-inv
dent-noninv

Discrete mutual information mi   dmi-mce
dmi-mje
dmi-jce

Continuous mutual information mi   cmi-mcde
cmi-mjde
cmi-jcde

Kullback-Leibler divergence kl kl-nonneg
kl-nonsymm
kl-conv
kl-inv
kl-ent
kl-dent

Estimation theory
Mean squared error mse   mse-bnv
Confidence interval ci     ci-wilks
Frequentist statistics
Likelihood theory lf
llf
mle
mll
mle-bias
Statistical hypotheses hyp
hyp-simp
hyp-point
hyp-tail

Hypothesis testing test
h0
h1
test-tail
tstat
size
power
alpha
cval
pval
pval-h0
Bayesian statistics
Probabilistic modeling gm
lf
prior
fpm
jl
post
map
ml
jl-lfnprior
post-jl
ml-jl
post-ind
Prior distributions prior-flat
prior-uni
prior-inf
prior-emp
prior-conj
prior-maxent
prior-eb
prior-ref

Bayesian inference eb
vb
bayes-th
bayes-rule