The tbles gies eressions for VR X nd ES X when X is n bsolutely continuous rndom rible secified by the stted df nd cdf. Eonentil Kumrswmy eonentil df cdf VR X ES X e b e e e b e e b /b / /b / Eonentited eonentil nerse eonentited eonentil eonentil Logistic eonentil e e e e eb Bb e e e e / e / e / / e b b e e / b / Eonentil etension e e / / Mrshll- Olkin eonentil Perks Berd e e eβ eβ eβ ρ ρ/β ρ eβ ρ/β e e eβ ρ ρ/β ρ eβ ρ/β β β ρ ρ ρ ρ/β β β β β ρ ρ ρ ρ/β Gomertz bη eb e η η eb e η η eb b η b η Gomertz Liner filure rte bη eb Bcd e dη e dη eb e η η eb c b e b / eηη eb c d b η c d e b b / b b η c d b b b Preto ck c c K K c K /c Kc /c c Kc c
Kumrswmy Preto F Preto Preto Preto ositie stble bck c c K c K c b K d B d d d d d d d d d /c k c k < k/c if c > > k/c if c < if c = K d d Bcd K c K ν ν e ν K c b K d d d d d d d /b / /c d d d d /c c k k c c K c d K e ν c d / e /ν K /b / /c d d d d d d k c kc cc k cc K / c d e /ν Gmm Kumrswmy gmm β eβ Γ bβ eβ γ β Γ γ β b Γ γβ Γ γ β Γ b β Q β Q β Q /b / β Q /b / Nkgmi Reflected gmm Comound Llce gmm Log gmm m m Γm m m e m Γ e < < β β < < r r Γr Q m m m Q m Q if Q if > β if β if > Qr Q if / Q if > / β / β if / β / β if > / e Q r m Q m Q if / / Q / Q if > / β / β/ if / β / β/ if > / e Q r
nerse gmm β eβ/ Γ Q β/ β β Q Q Stcy c cγ e/ c cγ Γγ Q γ c Q γ /c Q γ /c Lindley e e W e W e Lindley e e e W / e W / e b Bb b b b Uniform b b b b b uniform hkc c k c h k c h k /c /h /c k /c k/c /c /h Power function Power function b b / / / b /b b /b b Log bet d c b Bb c d b c d c b d c c dc b c dc b Comlementry bet B b b b b b b b Libby- Noick bet b Bb b b b b b b McDonld- Richrds bet r b r r b b r b Bb b /r r b br b/r /r b b /r b /r bet Arcsine c d b Bbdc b c d π b b c b dc π rcsin b c d c b sin π dc b c b b sin π 3
Tringulr bet if < bc if c b bbc if c < b if b < c c c b Bb if < bc if c b bbc if c < b if b < c b b c if < < c b b b b c if c b < 3 b c if < < c b b c cb 3 /c b b b c 3/ 3 if c b < b /c nerse bet Bb b b b b b b inerse bet Two sided ower c Bcd cd if < if < < Kumrswmy b b Norml Kumrswmy norml µ < < b µ Φ µ Φ µ < < b / c d cd cd if < if < < / if < / if < < b /b / µ Φ Φ µ b / cd cd / µ Φ µ µ Φ /b / if < / if < < /b / Φ Φ µ /b / µ / Q Eonentil ower / Γ / e µ < < Q µ if µ Q µ if > µ µ / Q / if / µ / Q / if > / / if / µ / / Q / / Q / / if > / 4
Skewed eonentil ower if K e K e if where K = / Γ/ Q if Q if Q if Q if > Q if Q Q Asymmetric eonentil ower norml Hlf norml Kumrswmy hlf norml Student s t K e if K e if where K = = µ Bb / Γ/ K K K Φ µ Φ b µ < < b Φ Φ b Γ n nπγ n n n < < if Q Q if Φ µ Q if b µ Φ Φ Φ Φ Φ b sign sign n n n Q if > / /b nsign if > Q if Q Q if > µ b Φ b n where = if < / = if / Φ Φ /b / n sign n where = if < / = if / 5
K ν ν Skewed Student s t Asymmetric Student s t Hlf Student s t ν if K ν ν ν if where Kν = Γν/ πνγν/ K ν ν ν if K ν ν ν if where Kν = Γν/ πνγν/ = Γ n nπγ n n n Kν Kν Kν min F ν F ν m where F ν is Student s t cdf min F ν m F ν where F ν is Student s t cdf n n Fν Fν where F min m is ν Student s t inerse cdf Fν min Fν m where Fν is Student s t inerse cdf n n n F ν min F ν m F ν min F ν m n n n Cuchy π µ < < π rctn µ µ tn π µ tn π Log Cuchy Hlf Cuchy Llce Poirud- Csno- Thoms- Agnn Llce π µ π e µ < < e if e if > π rctn µ π rctn µ e if < µ e µ if µ e if e if > e µ tn π π tn µ if < / µ if / if if > eµ e tn π tn π µ if < / µ if / if if > 6
Holl- Bhttchry Llce McGill Llce Log Llce Asymmetric Llce e if e if > ψ e ψ if e if > β β δ β β if δ βδ β if > δ κ τ κ e κ τ if κ τ κ e κτ if < e e if e e if > e ψ if e if > β δ β β if δ βδ β if > δ κ e κ τ if κ κ e κτ if < if if > ψ if / if > / δ β if /β β δ β / if > β τ κ if κτ if < κ κ κ κ κ κ if if > ψ if / ψ if > / β /β if β δβ β δ /ββ / β / δ β/ δ / / β if > β τ κ κ τ κ κ κ κ τκ κ κ κ κ τ κ 4 κ κ τ κ τ κ if κ κ κτ κ κτ if < κ κ 7
/ δ Asymmetric ower δ / Γ / e δ if δ / Γ / e δ if δ / if δ / if / δ / if / δ / if > / if / δ / / δ Logistic Hyerbolic secnt istic istic istic V e µ e µ < < sech π < < e e µ µ < < B e µ µ e < < B e µ e µ < < e µ e π rctn π e e e µ µ µ µ π tn µ π µ µ / µ / if > µ π tn π / µ µ Hlf istic Logistic Kumrswmy istic Eonentited istic e e ββ β β β bβ β β β β b β β β /β e/β e/β < < e e β β β b β β β /β B β β e/β β /b / /β / β /b / /β / Hosking istic norml k/k k /k < /k if k > > /k if k < < < if k = µ k /k k µ Φ k k k B k k eµ e µ Φ e Φ 8
norml Bb µ Φ µ Φ b µ e µ Φ µ b Φ b eµ e Φ b Burr Burr Burr X b b b b / b bd Bcd bd b cd / kc c c k / b /b /b B /b /b / b c d c d /b c k /k c d /b c d /b /c /c c d /b /k Kumrswmy Burr X Burr X Dgum Lom Lom Gumbel Kumrswmy Gumbel Gumbel Gumbel bkc c c k c k c k b kc c Bb k c c bk cb c b c Bb b e e e < < b e e e e e b < < e Bb e e e e b < < b e b c k b c k b /b / /k /c /k b /c b c b /c / / /b /k /c b /k /c b /c / / / / b / b b / e e e e b < < e e b e b /b / /b / b b / / b b / / Gumbel b Bcd e bd e b c c d b / e b c d / b / / c d 9
Fréchet Fréchet Kumrswmy Logistic Ryleigh Mrshll- Olkin e Bb e e e b b e e e b e / e / e / β β β e β e Bb e b e β e e / b e e b e / e / e e β β b / / /b / / / /β e b / b Γ / / b γ / / /b / / / /β / b /c Double c c e < < c e if e if > c c /c if / /c if > / if / / /c / Eonentited ower c c c e / c e / c e e / c / /c /c if > / / /c e / / / /
inerse b bc b e c b e c b c /b /b c /b /b generlized Ryleigh Chen BbΓ e γ Γ Q b β β e β e e β b Q β e e Q b Q /β / b /β Xie β β e / β e e e / β e e e / β /β /β Tukey Lmbd Goindrjulu β β β β β β β β Rmberg- Schmeiser β γ δ Freimer β β γ γ β δβ γ δγ γ β β ββ γ γγ Hnkin- Lee C β C B β n Stden- Loots 3 4 3 4 3 4 4 3 4 4 3 4 3 4 3 4 4 4 3 4 4 3 4 4 4 n Stden- King β δ βδ βδ β δ β δ βδ Log e e / / / Eonentil rithmic β eβ eβ eβ β β Eonentil geometric e e e e
Eonentil Poisson To- Leone Qudrtic β eβ eβ e b b β b = b 3 β = b e eβ e b β e 3 β 3 β 3 β 3 β 3 /3 β e /b b B /b b 3 β 4 3 β 3 4/3 β 4 Schbe Birnbum Sunders γ γ/ γ / / / γ / / γ γ γ Φ / / γ γ γ 4 γφ 4 γ Φ γ γγ γ γ γφ 4 4 γ Φ etreme lue µ /ξ ξ e /ξ ξ µ µ /ξ if ξ > µ /ξ if ξ < < < if ξ = e /ξ ξ µ µ ξ µ ξ ξ ξ ξ ξ