Index: The Book of Statistical Proofs ▷ Probability Distributions

Proofs and Definitions on Probability Distributions in the StatProofBook, as of 2023-12-23.

Distribution Def PDF CDF QF MGF Mean Med Mode Var Ent KL Marg Cond Other
Univariate
discrete
distributions		                            
Discrete uniform distribution duni duni-pmf duni-cdf duni-qf           duni-ent
duni-maxent		 duni-kl      
Bernoulli distribution bern bern-pmf       bern-mean     bern-var
bern-varrange		 bern-ent bern-kl      
Binomial distribution bin bin-pmf       bin-mean     bin-var
bin-varrange		 bin-ent bin-kl bin-margcond   bin-pgf
Beta-binomial distribution betabin betabin-pmf
betabin-pmfitogf		 betabin-cdf     bin-mean     bin-var
bin-varrange		 bin-ent        
Poisson distribution poiss poiss-pmf       poiss-mean     poiss-var          
Multivariate
discrete
distributions		                            
Categorical distribution cat cat-pmf       cat-mean     cat-cov cat-ent        
Multinomial distribution mult mult-pmf       mult-mean     mult-cov mult-ent        
Univariate
continuous
distributions		                            
Continuous uniform distribution cuni cuni-pdf cuni-cdf cuni-qf   cuni-mean cuni-med cuni-mode cuni-var cuni-dent
cuni-maxent		 cuni-kl      
Normal distribution norm
snorm		 norm-pdf norm-cdf
norm-cdfwerf		 norm-qf norm-mgf norm-mean norm-med norm-mode norm-var norm-dent
norm-maxent		 norm-kl     norm-mvn
norm-snorm
norm-chi2
norm-t
norm-lincomb
norm-gi
norm-probstd
norm-fwhm
norm-extr
norm-infl
t-distribution t
nst		 t-pdf                       t-mvt
nst-t
Gamma distribution gam
sgam		 gam-pdf gam-cdf gam-qf   gam-mean
gam-logmean
gam-xlogx		     gam-var gam-dent gam-kl     gam-sgam
gam-wish
gam-scal
Exponential distribution exp exp-pdf exp-cdf exp-qf exp-mgf exp-mean exp-med exp-mode exp-var         exp-gam
exp-skew
Log-normal distribution lognorm lognorm-pdf lognorm-cdf lognorm-qf   lognorm-mean lognorm-med lognorm-mode lognorm-var          
Chi-squared distribution chi2 chi2-pdf                       chi2-gam
chi2-mom
F-distribution f f-pdf                        
Beta distribution beta beta-pdf beta-cdf   beta-mgf beta-mean     beta-var         beta-chi2
Wald distribution wald wald-pdf     wald-mgf wald-mean     wald-var         wald-skew
wald-mome
ex-Gaussian distribution exg exg-pdf     exg-mgf exg-mean     exg-var         exg-skew
exg-mome
Multivariate
continuous
distributions		                            
Multivariate normal distribution mvn
bvn		 mvn-pdf
bvn-pdf
bvn-pdfcorr		       mvn-mean     mvn-cov mvn-dent mvn-kl mvn-marg mvn-cond mvn-matn
mvn-chi2
mvn-ltt
mvn-ind
mvn-indprod
Multivariate t-distribution mvt mvt-pdf                       mvt-f
Normal-gamma distribution ng ng-pdf       ng-mean     ng-cov ng-dent ng-kl ng-marg ng-cond ng-nw
ng-samp
Dirichlet distribution dir dir-pdf                 dir-kl     dir-ep
Matrix-variate
continuous
distributions		                            
Matrix-normal distribution matn matn-pdf       matn-mean     matn-cov matn-dent matn-kl     matn-mvn
matn-ltt
matn-trans
Wishart distribution wish                   wish-kl      
Normal-Wishart distribution nw nw-pdf       nw-mean