| Dystrybucja |
Funkcja rozkładu prawdopodobieństwa |
Entropia
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| Ciągłe jednolite prawo |
fa(x)=1b-w{\ Displaystyle f (x) = {\ Frac {1} {ba}}} dla w≤x≤b{\ displaystyle a \ leq x \ leq b}
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ln(b-w){\ Displaystyle \ ln (ba) \,}
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| Normalne prawo |
fa(x)=12πσ2exp(-(x-μ)22σ2){\ Displaystyle f (x) = {\ Frac {1} {\ sqrt {2 \ pi \ sigma ^ {2}}}} \ exp \ lewo (- {\ Frac {(x- \ mu) ^ {2} } {2 \ sigma ^ {2}}} \ right)} |
ln(σ2πmi){\ Displaystyle \ ln \ lewo (\ sigma {\ sqrt {2 \, \ pi \, e}} \ prawej)}
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| Prawo wykładnicze |
fa(x)=λexp(-λx){\ Displaystyle f (x) = \ lambda \ exp \ lewo (- \ lambda x \ prawej)} |
1-lnλ{\ Displaystyle 1- \ ln \ lambda \,}
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| Prawo Cauchy'ego |
fa(x)=λπ1λ2+x2{\ Displaystyle f (x) = {\ Frac {\ lambda} {\ pi}} {\ Frac {1} {\ lambda ^ {2} + x ^ {2}}}} |
ln(4πλ){\ Displaystyle \ ln (4 \ pi \ lambda) \,}
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| Prawo χ² |
fa(x)=12nie/2σnieΓ(nie/2)xnie2-1exp(-x2σ2){\ Displaystyle f (x) = {\ Frac {1} {2 ^ {n / 2} \ sigma ^ {n} \ Gamma (n / 2)}} x ^ {{\ Frac {n} {2}} -1} \ exp \ left (- {\ frac {x} {2 \ sigma ^ {2}}} \ right)} |
ln2σ2Γ(nie2)-(1-nie2)ψ(nie2)+nie2{\ Displaystyle \ ln 2 \ sigma ^ {2} \ Gamma \ lewo ({\ Frac {n} {2}} \ prawo) - \ lewo (1 - {\ Frac {n} {2}} \ prawo) \ psi \ left ({\ frac {n} {2}} \ right) + {\ frac {n} {2}}}
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| Rozkład gamma |
fa(x)=xα-1exp(-xβ)βαΓ(α){\ Displaystyle f (x) = {\ Frac {x ^ {\ alfa -1} \ exp (- {\ Frac {x} {\ beta}})} {\ beta ^ {\ alfa} \ Gamma (\ alfa )}}} |
ln(βΓ(α))+(1-α)ψ(α)+α{\ Displaystyle \ ln (\ beta \ Gamma (\ alfa)) + (1- \ alfa) \ psi (\ alfa) + \ alfa \,}
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| Prawo logistyczne |
fa(x)=mi-x(1+mi-x)2{\ Displaystyle f (x) = {\ Frac {e ^ {- x}} {(1 + e ^ {- x}) ^ {2}}}} |
2{\ Displaystyle 2 \,}
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| Statystyka Maxwella-Boltzmanna |
fa(x)=4π-12β32x2exp(-βx2){\ Displaystyle f (x) = 4 \ pi ^ {- {\ Frac {1} {2}}} \ beta ^ {\ Frac {3} {2}} x ^ {2} \ exp (- \ beta x ^ {2})} |
12lnπβ+γ-1/2{\ Displaystyle {\ Frac {1} {2}} \ ln {\ Frac {\ pi} {\ beta}} + \ gamma -1/2}
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| Dystrybucja Pareto |
fa(x)=wkwxw+1{\ Displaystyle f (x) = {\ Frac {ak ^ {a}} {x ^ {a + 1}}}} |
lnkw+1+1w{\ Displaystyle \ ln {\ Frac {k} {a}} + 1 + {\ Frac {1} {a}}}
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| Prawo studenta |
fa(x)=(1+x2/nie)-nie+12nieb(12,nie2){\ Displaystyle f (x) = {\ Frac {(1 + x ^ {2} / n) ^ {- {\ Frac {n + 1} {2}}}} {{\ sqrt {n}} B ( {\ frac {1} {2}}, {\ frac {n} {2}})}}} |
nie+12ψ(nie+12)-ψ(nie2)+lnnieb(12,nie2){\ Displaystyle {\ Frac {n + 1} {2}} \ psi \ lewo ({\ Frac {n + 1} {2}} \ prawo) - \ psi \ lewo ({\ Frac {n} {2} } \ right) + \ ln {\ sqrt {n}} B \ left ({\ frac {1} {2}}, {\ frac {n} {2}} \ right)}
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| Rozkład Weibulla |
fa(x)=vsαxvs-1exp(-xvsα){\ Displaystyle f (x) = {\ Frac {c} {\ alfa}} x ^ {c-1} \ exp \ lewo (- {\ Frac {x ^ {c}} {\ alfa}} \ prawej) } |
(vs-1)γvs+lnα1/vsvs+1{\ Displaystyle {\ Frac {(c-1) \ gamma} {c}} + \ ln {\ Frac {\ alpha ^ {1 / c}} {c}} + 1}
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| Wielowymiarowe prawo normalne |
faX(x1,...,xNIE)={\ displaystyle f_ {X} (x_ {1}, \ kropki, x_ {N}) =} 1(2π)NIE/2|Σ|1/2exp(-12(x-μ)⊤Σ-1(x-μ)){\ Displaystyle {\ Frac {1} {(2 \ pi) ^ {N / 2} \ lewo | \ Sigma \ prawo | ^ {1/2}}} \ exp \ lewo (- {\ Frac {1} { 2}} (x- \ mu) ^ {\ top} \ Sigma ^ {- 1} (x- \ mu) \ right)}
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12ln{(2πmi)NIE|Σ|}{\ Displaystyle {\ Frac {1} {2}} \ ln \ {(2 \ pi e) ^ {N} | \ Sigma | \}}
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