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Probability distribution
In p r o b a b i l i t y t h e o r y a n d s t a t i s t i c s , t h e L a p l a c e d i s t r i b u t i o n i s a c o n t i n u o u s p r o b a b i l i t y d i s t r i b u t i o n n a m e d a f t e r P i e r r e - S i m o n L a p l a c e . I t i s a l s o s o m e t i m e s c a l l e d t h e d o u b l e e x p o n e n t i a l d i s t r i b u t i o n , b e c a u s e i t c a n b e t h o u g h t o f a s t w o e x p o n e n t i a l d i s t r i b u t i o n s ( w i t h a n a d d i t i o n a l l o c a t i o n p a r a m e t e r ) s p l i c e d t o g e t h e r a l o n g t h e a b s c i s s a , a l t h o u g h t h e t e r m i s a l s o s o m e t i m e s u s e d t o r e f e r t o t h e G u m b e l d i s t r i b u t i o n . T h e d i f f e r e n c e b e t w e e n t w o i n d e p e n d e n t i d e n t i c a l l y d i s t r i b u t e d e x p o n e n t i a l r a n d o m v a r i a b l e s i s g o v e r n e d b y a L a p l a c e d i s t r i b u t i o n , a s i s a B r o w n i a n m o t i o n e v a l u a t e d a t a n e x p o n e n t i a l l y d i s t r i b u t e d r a n d o m t i m e [ c i t a t i o n n e e d e d ] . I n c r e m e n t s o f L a p l a c e m o t i o n o r a v a r i a n c e g a m m a p r o c e s s e v a l u a t e d o v e r t h e t i m e s c a l e a l s o h a v e a L a p l a c e d i s t r i b u t i o n .
D e f i n i t i o n s [ e d i t ]
P r o b a b i l i t y d e n s i t y f u n c t i o n [ e d i t ]
A r a n d o m v a r i a b l e h a s a
Laplace
(
μ
,
b
)
{\displaystyle \operatorname {Laplace} (\mu ,b)}
d i s t r i b u t i o n i f i t s p r o b a b i l i t y d e n s i t y f u n c t i o n is
f
(
x
∣
μ
,
b
)
=
1
2
b
exp
(
−
|
x
−
μ
|
b
)
,
{\displaystyle f(x\mid \mu ,b)={\frac {1}{2b}}\exp \left(-{\frac {|x-\mu |}{b}}\right),}
w h e r e
μ
{\displaystyle \mu }
i s a l o c a t i o n p a r a m e t e r , a n d
b
>
0
{\displaystyle b>0}
, w h i c h i s s o m e t i m e s r e f e r r e d t o a s t h e " d i v e r s i t y " , i s a s c a l e p a r a m e t e r . I f
μ
=
0
{\displaystyle \mu =0}
a n d
b
=
1
{\displaystyle b=1}
, t h e p o s i t i v e h a l f - l i n e i s e x a c t l y a n e x p o n e n t i a l d i s t r i b u t i o n s c a l e d b y 1 / 2 .
T h e p r o b a b i l i t y d e n s i t y f u n c t i o n o f t h e L a p l a c e d i s t r i b u t i o n i s a l s o r e m i n i s c e n t o f t h e n o r m a l d i s t r i b u t i o n ; h o w e v e r , w h e r e a s t h e n o r m a l d i s t r i b u t i o n i s e x p r e s s e d i n t e r m s o f t h e s q u a r e d d i f f e r e n c e f r o m t h e m e a n
μ
{\displaystyle \mu }
, t h e L a p l a c e d e n s i t y i s e x p r e s s e d i n t e r m s o f t h e a b s o l u t e d i f f e r e n c e f r o m t h e m e a n . C o n s e q u e n t l y , t h e L a p l a c e d i s t r i b u t i o n h a s f a t t e r t a i l s t h a n t h e n o r m a l d i s t r i b u t i o n . I t i s a s p e c i a l c a s e o f t h e g e n e r a l i z e d n o r m a l d i s t r i b u t i o n a n d t h e h y p e r b o l i c d i s t r i b u t i o n . C o n t i n u o u s s y m m e t r i c d i s t r i b u t i o n s t h a t h a v e e x p o n e n t i a l t a i l s , l i k e t h e L a p l a c e d i s t r i b u t i o n , b u t w h i c h h a v e p r o b a b i l i t y d e n s i t y f u n c t i o n s t h a t a r e d i f f e r e n t i a b l e a t t h e m o d e i n c l u d e t h e l o g i s t i c d i s t r i b u t i o n , h y p e r b o l i c s e c a n t d i s t r i b u t i o n , a n d t h e C h a m p e r n o w n e d i s t r i b u t i o n .
C u m u l a t i v e d i s t r i b u t i o n f u n c t i o n [ e d i t ]
T h e L a p l a c e d i s t r i b u t i o n i s e a s y t o i n t e g r a t e ( i f o n e d i s t i n g u i s h e s t w o s y m m e t r i c c a s e s ) d u e t o t h e u s e o f t h e a b s o l u t e v a l u e f u n c t i o n . I t s c u m u l a t i v e d i s t r i b u t i o n f u n c t i o n i s a s f o l l o w s :
F
(
x
)
=
∫
−
∞
x
f
(
u
)
d
u
=
{
1
2
exp
(
x
−
μ
b
)
if
x
<
μ
1
−
1
2
exp
(
−
x
−
μ
b
)
if
x
≥
μ
=
1
2
+
1
2
sgn
(
x
−
μ
)
(
1
−
exp
(
−
|
x
−
μ
|
b
)
)
.
{\displaystyle {\begin{aligned}F(x )&=\int _{-\infty }^{x}\!\!f(u )\,\mathrm {d} u={\begin{cases}{\frac {1}{2}}\exp \left({\frac {x-\mu }{b}}\right)&{\mbox{if }}x<\mu \\1-{\frac {1}{2}}\exp \left(-{\frac {x-\mu }{b}}\right)&{\mbox{if }}x\geq \mu \end{cases}}\\&={\tfrac {1}{2}}+{\tfrac {1}{2}}\operatorname {sgn}(x-\mu )\left(1-\exp \left(-{\frac {|x-\mu |}{b}}\right)\right).\end{aligned}}}
T h e i n v e r s e c u m u l a t i v e d i s t r i b u t i o n f u n c t i o n i s g i v e n b y
F
−
1
(
p
)
=
μ
−
b
sgn
(
p
−
0.5
)
ln
(
1
−
2
|
p
−
0.5
|
)
.
{\displaystyle F^{-1}(p )=\mu -b\,\operatorname {sgn}(p-0.5)\,\ln(1-2|p-0.5|).}
P r o p e r t i e s [ e d i t ]
M o m e n t s [ e d i t ]
μ
r
′
=
(
1
2
)
∑
k
=
0
r
[
r
!
(
r
−
k
)
!
b
k
μ
(
r
−
k
)
{
1
+
(
−
1
)
k
}
]
.
{\displaystyle \mu _{r}'={\bigg (}{\frac {1}{2}}{\bigg )}\sum _{k=0}^{r}{\bigg [}{\frac {r!}{(r-k)!}}b^{k}\mu ^{(r-k)}\{1+(-1)^{k}\}{\bigg ]}.}
R e l a t e d d i s t r i b u t i o n s [ e d i t ]
● If
X
∼
Laplace
(
μ
,
b
)
{\displaystyle X\sim {\textrm {Laplace}}(\mu ,b)}
t h e n
k
X
+
c
∼
Laplace
(
k
μ
+
c
,
|
k
|
b
)
{\displaystyle kX+c\sim {\textrm {Laplace}}(k\mu +c,|k|b)}
.
● If
X
∼
Laplace
(
0
,
1
)
{\displaystyle X\sim {\textrm {Laplace}}(0,1)}
t h e n
b
X
∼
Laplace
(
0
,
b
)
{\displaystyle bX\sim {\textrm {Laplace}}(0,b)}
.
● If
X
∼
Laplace
(
0
,
b
)
{\displaystyle X\sim {\textrm {Laplace}}(0,b)}
t h e n
|
X
|
∼
Exponential
(
b
−
1
)
{\displaystyle \left|X\right|\sim {\textrm {Exponential}}\left(b^{-1}\right)}
( e x p o n e n t i a l d i s t r i b u t i o n ) .
● If
X
,
Y
∼
Exponential
(
λ
)
{\displaystyle X,Y\sim {\textrm {Exponential}}(\lambda )}
t h e n
X
−
Y
∼
Laplace
(
0
,
λ
−
1
)
{\displaystyle X-Y\sim {\textrm {Laplace}}\left(0,\lambda ^{-1}\right)}
.
● If
X
∼
Laplace
(
μ
,
b
)
{\displaystyle X\sim {\textrm {Laplace}}(\mu ,b)}
t h e n
|
X
−
μ
|
∼
Exponential
(
b
−
1
)
{\displaystyle \left|X-\mu \right|\sim {\textrm {Exponential}}(b^{-1})}
.
● If
X
∼
Laplace
(
μ
,
b
)
{\displaystyle X\sim {\textrm {Laplace}}(\mu ,b)}
t h e n
X
∼
EPD
(
μ
,
b
,
1
)
{\displaystyle X\sim {\textrm {EPD}}(\mu ,b,1)}
( e x p o n e n t i a l p o w e r d i s t r i b u t i o n ) .
● If
X
1
,
.
.
.
,
X
4
∼
N
(
0
,
1
)
{\displaystyle X_{1},...,X_{4}\sim {\textrm {N}}(0,1)}
( n o r m a l d i s t r i b u t i o n ) t h e n
X
1
X
2
−
X
3
X
4
∼
Laplace
(
0
,
1
)
{\displaystyle X_{1}X_{2}-X_{3}X_{4}\sim {\textrm {Laplace}}(0,1)}
a n d
(
X
1
2
−
X
2
2
+
X
3
2
−
X
4
2
)
/
2
∼
Laplace
(
0
,
1
)
{\displaystyle (X_{1}^{2}-X_{2}^{2}+X_{3}^{2}-X_{4}^{2})/2\sim {\textrm {Laplace}}(0,1)}
.
● If
X
i
∼
Laplace
(
μ
,
b
)
{\displaystyle X_{i}\sim {\textrm {Laplace}}(\mu ,b)}
t h e n
2
b
∑
i
=
1
n
|
X
i
−
μ
|
∼
χ
2
(
2
n
)
{\displaystyle {\frac {\displaystyle 2}{b}}\sum _{i=1}^{n}|X_{i}-\mu |\sim \chi ^{2}(2n)}
( c h i - s q u a r e d d i s t r i b u t i o n ) .
● If
X
,
Y
∼
Laplace
(
μ
,
b
)
{\displaystyle X,Y\sim {\textrm {Laplace}}(\mu ,b)}
t h e n
|
X
−
μ
|
|
Y
−
μ
|
∼
F
(
2
,
2
)
{\displaystyle {\tfrac {|X-\mu |}{|Y-\mu |}}\sim \operatorname {F} (2,2)}
. ( F - d i s t r i b u t i o n )
● If
X
,
Y
∼
U
(
0
,
1
)
{\displaystyle X,Y\sim {\textrm {U}}(0,1)}
( u n i f o r m d i s t r i b u t i o n ) t h e n
log
(
X
/
Y
)
∼
Laplace
(
0
,
1
)
{\displaystyle \log(X/Y)\sim {\textrm {Laplace}}(0,1)}
.
● If
X
∼
Exponential
(
λ
)
{\displaystyle X\sim {\textrm {Exponential}}(\lambda )}
a n d
Y
∼
Bernoulli
(
0.5
)
{\displaystyle Y\sim {\textrm {Bernoulli}}(0.5)}
( B e r n o u l l i d i s t r i b u t i o n ) i n d e p e n d e n t o f
X
{\displaystyle X}
, t h e n
X
(
2
Y
−
1
)
∼
Laplace
(
0
,
λ
−
1
)
{\displaystyle X(2Y-1)\sim {\textrm {Laplace}}\left(0,\lambda ^{-1}\right)}
.
● If
X
∼
Exponential
(
λ
)
{\displaystyle X\sim {\textrm {Exponential}}(\lambda )}
a n d
Y
∼
Exponential
(
ν
)
{\displaystyle Y\sim {\textrm {Exponential}}(\nu )}
i n d e p e n d e n t o f
X
{\displaystyle X}
, t h e n
λ
X
−
ν
Y
∼
Laplace
(
0
,
1
)
{\displaystyle \lambda X-\nu Y\sim {\textrm {Laplace}}(0,1)}
.
● If
X
{\displaystyle X}
h a s a R a d e m a c h e r d i s t r i b u t i o n a n d
Y
∼
Exponential
(
λ
)
{\displaystyle Y\sim {\textrm {Exponential}}(\lambda )}
t h e n
X
Y
∼
Laplace
(
0
,
1
/
λ
)
{\displaystyle XY\sim {\textrm {Laplace}}(0,1/\lambda )}
.
● If
V
∼
Exponential
(
1
)
{\displaystyle V\sim {\textrm {Exponential}}(1 )}
a n d
Z
∼
N
(
0
,
1
)
{\displaystyle Z\sim N(0,1)}
i n d e p e n d e n t o f
V
{\displaystyle V}
, t h e n
X
=
μ
+
b
2
V
Z
∼
L
a
p
l
a
c
e
(
μ
,
b
)
{\displaystyle X=\mu +b{\sqrt {2V}}Z\sim \mathrm {Laplace} (\mu ,b)}
.
● If
X
∼
GeometricStable
(
2
,
0
,
λ
,
0
)
{\displaystyle X\sim {\textrm {GeometricStable}}(2,0,\lambda ,0)}
( g e o m e t r i c s t a b l e d i s t r i b u t i o n ) t h e n
X
∼
Laplace
(
0
,
λ
)
{\displaystyle X\sim {\textrm {Laplace}}(0,\lambda )}
.
● T h e L a p l a c e d i s t r i b u t i o n i s a l i m i t i n g c a s e o f t h e h y p e r b o l i c d i s t r i b u t i o n .
● If
X
|
Y
∼
N
(
μ
,
Y
2
)
{\displaystyle X|Y\sim {\textrm {N}}(\mu ,Y^{2})}
w i t h
Y
∼
Rayleigh
(
b
)
{\displaystyle Y\sim {\textrm {Rayleigh}}(b )}
( R a y l e i g h d i s t r i b u t i o n ) t h e n
X
∼
Laplace
(
μ
,
b
)
{\displaystyle X\sim {\textrm {Laplace}}(\mu ,b)}
. N o t e t h a t i f
Y
∼
Rayleigh
(
b
)
{\displaystyle Y\sim {\textrm {Rayleigh}}(b )}
, t h e n
Y
2
∼
Gamma
(
1
,
2
b
2
)
{\displaystyle Y^{2}\sim {\textrm {Gamma}}(1,2b^{2})}
w i t h
E
(
Y
2
)
=
2
b
2
{\displaystyle {\textrm {E}}(Y^{2})=2b^{2}}
, w h i c h i n t u r n e q u a l s t h e e x p o n e n t i a l d i s t r i b u t i o n
Exp
(
1
/
(
2
b
2
)
)
{\displaystyle {\textrm {Exp}}(1/(2b^{2}))}
.
● G i v e n a n i n t e g e r
n
≥
1
{\displaystyle n\geq 1}
, i f
X
i
,
Y
i
∼
Γ
(
1
n
,
b
)
{\displaystyle X_{i},Y_{i}\sim \Gamma \left({\frac {1}{n}},b\right)}
( g a m m a d i s t r i b u t i o n , u s i n g
k
,
θ
{\displaystyle k,\theta }
c h a r a c t e r i z a t i o n ) , t h e n
∑
i
=
1
n
(
μ
n
+
X
i
−
Y
i
)
∼
Laplace
(
μ
,
b
)
{\displaystyle \sum _{i=1}^{n}\left({\frac {\mu }{n}}+X_{i}-Y_{i}\right)\sim {\textrm {Laplace}}(\mu ,b)}
( i n f i n i t e d i v i s i b i l i t y ) [ 2 ]
● If X h a s a L a p l a c e d i s t r i b u t i o n , t h e n Y = e X h a s a l o g - L a p l a c e d i s t r i b u t i o n ; c o n v e r s e l y , i f X h a s a l o g - L a p l a c e d i s t r i b u t i o n , t h e n i t s l o g a r i t h m h a s a L a p l a c e d i s t r i b u t i o n .
P r o b a b i l i t y o f a L a p l a c e b e i n g g r e a t e r t h a n a n o t h e r [ e d i t ]
L e t
X
,
Y
{\displaystyle X,Y}
b e i n d e p e n d e n t l a p l a c e r a n d o m v a r i a b l e s :
X
∼
Laplace
(
μ
X
,
b
X
)
{\displaystyle X\sim {\textrm {Laplace}}(\mu _{X},b_{X})}
a n d
Y
∼
Laplace
(
μ
Y
,
b
Y
)
{\displaystyle Y\sim {\textrm {Laplace}}(\mu _{Y},b_{Y})}
, a n d w e w a n t t o c o m p u t e
P
(
X
>
Y
)
{\displaystyle P(X>Y )}
.
T h e p r o b a b i l i t y o f
P
(
X
>
Y
)
{\displaystyle P(X>Y )}
c a n b e r e d u c e d ( u s i n g t h e p r o p e r t i e s b e l o w ) t o
P
(
μ
+
b
Z
1
>
Z
2
)
{\displaystyle P(\mu +bZ_{1}>Z_{2})}
, w h e r e
Z
1
,
Z
2
∼
Laplace
(
0
,
1
)
{\displaystyle Z_{1},Z_{2}\sim {\textrm {Laplace}}(0,1)}
. T h i s p r o b a b i l i t y i s e q u a l t o
P
(
μ
+
b
Z
1
>
Z
2
)
=
{
b
2
e
μ
/
b
−
e
μ
2
(
b
2
−
1
)
,
when
μ
<
0
1
−
b
2
e
−
μ
/
b
−
e
−
μ
2
(
b
2
−
1
)
,
when
μ
>
0
{\displaystyle P(\mu +bZ_{1}>Z_{2})={\begin{cases}{\frac {b^{2}e^{\mu /b}-e^{\mu }}{2(b^{2}-1)}},&{\text{when }}\mu <0\\1-{\frac {b^{2}e^{-\mu /b}-e^{-\mu }}{2(b^{2}-1)}},&{\text{when }}\mu >0\\\end{cases}}}
W h e n
b
=
1
{\displaystyle b=1}
, b o t h e x p r e s s i o n s a r e r e p l a c e d b y t h e i r l i m i t a s
b
→
1
{\displaystyle b\to 1}
:
P
(
μ
+
Z
1
>
Z
2
)
=
{
e
μ
(
2
−
μ
)
4
,
when
μ
<
0
1
−
e
−
μ
(
2
+
μ
)
4
,
when
μ
>
0
{\displaystyle P(\mu +Z_{1}>Z_{2})={\begin{cases}e^{\mu }{\frac {(2-\mu )}{4}},&{\text{when }}\mu <0\\1-e^{-\mu }{\frac {(2+\mu )}{4}},&{\text{when }}\mu >0\\\end{cases}}}
T o c o m p u t e t h e c a s e f o r
μ
>
0
{\displaystyle \mu >0}
, n o t e t h a t
P
(
μ
+
Z
1
>
Z
2
)
=
1
−
P
(
μ
+
Z
1
<
Z
2
)
=
1
−
P
(
−
μ
−
Z
1
>
−
Z
2
)
=
1
−
P
(
−
μ
+
Z
1
>
Z
2
)
{\displaystyle P(\mu +Z_{1}>Z_{2})=1-P(\mu +Z_{1}<Z_{2})=1-P(-\mu -Z_{1}>-Z_{2})=1-P(-\mu +Z_{1}>Z_{2})}
s i n c e
Z
∼
−
Z
{\displaystyle Z\sim -Z}
w h e n
Z
∼
Laplace
(
0
,
1
)
{\displaystyle Z\sim {\textrm {Laplace}}(0,1)}
R e l a t i o n t o t h e e x p o n e n t i a l d i s t r i b u t i o n [ e d i t ]
A L a p l a c e r a n d o m v a r i a b l e c a n b e r e p r e s e n t e d a s t h e d i f f e r e n c e o f t w o i n d e p e n d e n t a n d i d e n t i c a l l y d i s t r i b u t e d ( i i d ) e x p o n e n t i a l r a n d o m v a r i a b l e s . [ 2 ] O n e w a y t o s h o w t h i s i s b y u s i n g t h e c h a r a c t e r i s t i c f u n c t i o n a p p r o a c h . F o r a n y s e t o f i n d e p e n d e n t c o n t i n u o u s r a n d o m v a r i a b l e s , f o r a n y l i n e a r c o m b i n a t i o n o f t h o s e v a r i a b l e s , i t s c h a r a c t e r i s t i c f u n c t i o n ( w h i c h u n i q u e l y d e t e r m i n e s t h e d i s t r i b u t i o n ) c a n b e a c q u i r e d b y m u l t i p l y i n g t h e c o r r e s p o n d i n g c h a r a c t e r i s t i c f u n c t i o n s .
C o n s i d e r t w o i . i . d r a n d o m v a r i a b l e s
X
,
Y
∼
Exponential
(
λ
)
{\displaystyle X,Y\sim {\textrm {Exponential}}(\lambda )}
. T h e c h a r a c t e r i s t i c f u n c t i o n s f o r
X
,
−
Y
{\displaystyle X,-Y}
a r e
λ
−
i
t
+
λ
,
λ
i
t
+
λ
{\displaystyle {\frac {\lambda }{-it+\lambda }},\quad {\frac {\lambda }{it+\lambda }}}
r e s p e c t i v e l y . O n m u l t i p l y i n g t h e s e c h a r a c t e r i s t i c f u n c t i o n s ( e q u i v a l e n t t o t h e c h a r a c t e r i s t i c f u n c t i o n o f t h e s u m o f t h e r a n d o m v a r i a b l e s
X
+
(
−
Y
)
{\displaystyle X+(-Y)}
) , t h e r e s u l t i s
λ
2
(
−
i
t
+
λ
)
(
i
t
+
λ
)
=
λ
2
t
2
+
λ
2
.
{\displaystyle {\frac {\lambda ^{2}}{(-it+\lambda )(it+\lambda )}}={\frac {\lambda ^{2}}{t^{2}+\lambda ^{2}}}.}
T h i s i s t h e s a m e a s t h e c h a r a c t e r i s t i c f u n c t i o n f o r
Z
∼
Laplace
(
0
,
1
/
λ
)
{\displaystyle Z\sim {\textrm {Laplace}}(0,1/\lambda )}
, w h i c h i s
1
1
+
t
2
λ
2
.
{\displaystyle {\frac {1}{1+{\frac {t^{2}}{\lambda ^{2}}}}}.}
S a r g a n d i s t r i b u t i o n s [ e d i t ]
S a r g a n d i s t r i b u t i o n s a r e a s y s t e m o f d i s t r i b u t i o n s o f w h i c h t h e L a p l a c e d i s t r i b u t i o n i s a c o r e m e m b e r . A
p
{\displaystyle p}
t h o r d e r S a r g a n d i s t r i b u t i o n h a s d e n s i t y [ 3 ] [ 4 ]
f
p
(
x
)
=
1
2
exp
(
−
α
|
x
|
)
1
+
∑
j
=
1
p
β
j
α
j
|
x
|
j
1
+
∑
j
=
1
p
j
!
β
j
,
{\displaystyle f_{p}(x )={\tfrac {1}{2}}\exp(-\alpha |x|){\frac {\displaystyle 1+\sum _{j=1}^{p}\beta _{j}\alpha ^{j}|x|^{j}}{\displaystyle 1+\sum _{j=1}^{p}j!\beta _{j}}},}
f o r p a r a m e t e r s
α
≥
0
,
β
j
≥
0
{\displaystyle \alpha \geq 0,\beta _{j}\geq 0}
. T h e L a p l a c e d i s t r i b u t i o n r e s u l t s f o r
p
=
0
{\displaystyle p=0}
.
S t a t i s t i c a l i n f e r e n c e [ e d i t ]
G i v e n
n
{\displaystyle n}
i n d e p e n d e n t a n d i d e n t i c a l l y d i s t r i b u t e d s a m p l e s
x
1
,
x
2
,
.
.
.
,
x
n
{\displaystyle x_{1},x_{2},...,x_{n}}
, t h e m a x i m u m l i k e l i h o o d ( M L E ) e s t i m a t o r o f
μ
{\displaystyle \mu }
i s t h e s a m p l e m e d i a n , [ 5 ]
μ
^
=
m
e
d
(
x
)
.
{\displaystyle {\hat {\mu }}=\mathrm {med} (x ).}
T h e M L E e s t i m a t o r o f
b
{\displaystyle b}
i s t h e m e a n a b s o l u t e d e v i a t i o n f r o m t h e m e d i a n , [ c i t a t i o n n e e d e d ]
b
^
=
1
n
∑
i
=
1
n
|
x
i
−
μ
^
|
.
{\displaystyle {\hat {b}}={\frac {1}{n}}\sum _{i=1}^{n}|x_{i}-{\hat {\mu }}|.}
r e v e a l i n g a l i n k b e t w e e n t h e L a p l a c e d i s t r i b u t i o n a n d l e a s t a b s o l u t e d e v i a t i o n s .
A c o r r e c t i o n f o r s m a l l s a m p l e s c a n b e a p p l i e d a s f o l l o w s :
b
^
∗
=
b
^
⋅
n
/
(
n
−
2
)
{\displaystyle {\hat {b}}^{*}={\hat {b}}\cdot n/(n-2)}
( s e e : e x p o n e n t i a l d i s t r i b u t i o n # P a r a m e t e r e s t i m a t i o n ) .
O c c u r r e n c e a n d a p p l i c a t i o n s [ e d i t ]
T h e L a p l a c i a n d i s t r i b u t i o n h a s b e e n u s e d i n s p e e c h r e c o g n i t i o n t o m o d e l p r i o r s o n D F T c o e f f i c i e n t s [ 6 ] a n d i n J P E G i m a g e c o m p r e s s i o n t o m o d e l A C c o e f f i c i e n t s [ 7 ] g e n e r a t e d b y a D C T .
The addition of noise drawn from a Laplacian distribution, with scaling parameter appropriate to a function's sensitivity, to the output of a statistical database query is the most common means to provide differential privacy in statistical databases.
Fitted Laplace distribution to maximum one-day rainfalls [8]
The Laplace distribution, being a composite or double distribution, is applicable in situations where the lower values originate under different external conditions than the higher ones so that they follow a different pattern.[12]
Random variate generation [ edit ]
Given a random variable
U
{\displaystyle U}
drawn from the uniform distribution in the interval
(
−
1
/
2
,
1
/
2
)
{\displaystyle \left(-1/2,1/2\right)}
, the random variable
X
=
μ
−
b
sgn
(
U
)
ln
(
1
−
2
|
U
|
)
{\displaystyle X=\mu -b\,\operatorname {sgn}(U)\,\ln(1-2|U|)}
has a Laplace distribution with parameters
μ
{\displaystyle \mu }
and
b
{\displaystyle b}
. This follows from the inverse cumulative distribution function given above.
A
Laplace
(
0
,
b
)
{\displaystyle {\textrm {Laplace}}(0,b)}
variate can also be generated as the difference of two i.i.d.
Exponential
(
1
/
b
)
{\displaystyle {\textrm {Exponential}}(1/b)}
random variables. Equivalently,
Laplace
(
0
,
1
)
{\displaystyle {\textrm {Laplace}}(0,1)}
can also be generated as the logarithm of the ratio of two i.i.d. uniform random variables.
History [ edit ]
This distribution is often referred to as "Laplace's first law of errors". He published it in 1774, modeling the frequency of an error as an exponential function of its magnitude once its sign was disregarded. Laplace would later replace this model with his "second law of errors", based on the normal distribution, after the discovery of the central limit theorem .[13] [14]
Keynes published a paper in 1911 based on his earlier thesis wherein he showed that the Laplace distribution minimised the absolute deviation from the median.[15]
See also [ edit ]
References [ edit ]
^ a b Norton, Matthew; Khokhlov, Valentyn; Uryasev, Stan (2019). "Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation" (PDF) . Annals of Operations Research . 299 (1–2). Springer: 1281–1315. doi :10.1007/s10479-019-03373-1 . Retrieved 2023-02-27 .
^ a b Kotz, Samuel; Kozubowski, Tomasz J.; Podgórski, Krzysztof (2001). The Laplace distribution and generalizations: a revisit with applications to Communications, Economics, Engineering and Finance . Birkhauser. pp. 23 (Proposition 2.2.2, Equation 2.2.8). ISBN 9780817641665 .
^ Everitt, B.S. (2002) The Cambridge Dictionary of Statistics , CUP. ISBN 0-521-81099-X
^ Johnson, N.L., Kotz S., Balakrishnan, N. (1994) Continuous Univariate Distributions , Wiley. ISBN 0-471-58495-9 . p. 60
^ Robert M. Norton (May 1984). "The Double Exponential Distribution: Using Calculus to Find a Maximum Likelihood Estimator". The American Statistician . 38 (2). American Statistical Association: 135–136. doi :10.2307/2683252 . JSTOR 2683252 .
^ Eltoft, T.; Taesu Kim; Te-Won Lee (2006). "On the multivariate Laplace distribution" (PDF) . IEEE Signal Processing Letters . 13 (5): 300–303. doi :10.1109/LSP.2006.870353 . S2CID 1011487 . Archived from the original (PDF) on 2013-06-06. Retrieved 2012-07-04 .
^ Minguillon, J.; Pujol, J. (2001). "JPEG standard uniform quantization error modeling with applications to sequential and progressive operation modes" (PDF) . Journal of Electronic Imaging . 10 (2): 475–485. doi :10.1117/1.1344592 . hdl :10609/6263 .
^ CumFreq for probability distribution fitting
^ Pardo, Scott (2020). Statistical Analysis of Empirical Data Methods for Applied Sciences . Springer. p. 58. ISBN 978-3-030-43327-7 .
^ Kou, S.G. (August 8, 2002). "A Jump-Diffusion Model for Option Pricing" . Management Science . 48 (8): 1086–1101. doi :10.1287/mnsc.48.8.1086.166 . JSTOR 822677 . Retrieved 2022-03-01 .
^ Chen, Jian (2018). General Equilibrium Option Pricing Method: Theoretical and Empirical Study . Springer. p. 70. ISBN 9789811074288 .
^ A collection of composite distributions
^ Laplace, P-S. (1774). Mémoire sur la probabilité des causes par les évènements. Mémoires de l’Academie Royale des Sciences Presentés par Divers Savan, 6, 621–656
^ Wilson, Edwin Bidwell (1923). "First and Second Laws of Error". Journal of the American Statistical Association . 18 (143). Informa UK Limited: 841–851. doi :10.1080/01621459.1923.10502116 . ISSN 0162-1459 . This article incorporates text from this source, which is in the public domain .
^ Keynes, J. M. (1911). "The Principal Averages and the Laws of Error which Lead to Them" . Journal of the Royal Statistical Society . 74 (3). JSTOR: 322–331. doi :10.2307/2340444 . ISSN 0952-8385 . JSTOR 2340444 .
External links [ edit ]