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A p p e a r a n c e
F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
In probability and statistics , the reciprocal distribution , also known as the log-uniform distribution , is a continuous probability distribution . It is characterised by its probability density function , within the support of the distribution, being proportional to the reciprocal of the variable.
The reciprocal distribution is an example of an inverse distribution , and the reciprocal (inverse) of a random variable with a reciprocal distribution itself has a reciprocal distribution.
Definition
[ edit ]
The probability density function (pdf) of the reciprocal distribution is
f
(
x
;
a
,
b
)
=
1
x
[
ln
(
b
)
−
ln
(
a
)
]
for
a
≤
x
≤
b
and
a
>
0.
{\displaystyle f(x;a,b)={\frac {1}{x[\ln(b )-\ln(a )]}}\quad {\text{ for }}a\leq x\leq b{\text{ and }}a>0.}
Here,
a
{\displaystyle a}
and
b
{\displaystyle b}
are the parameters of the distribution, which are the lower and upper bounds of the support , and
ln
{\displaystyle \ln }
is the natural log . The cumulative distribution function is
F
(
x
;
a
,
b
)
=
ln
(
x
)
−
ln
(
a
)
ln
(
b
)
−
ln
(
a
)
for
a
≤
x
≤
b
.
{\displaystyle F(x;a,b)={\frac {\ln(x )-\ln(a )}{\ln(b )-\ln(a )}}\quad {\text{ for }}a\leq x\leq b.}
Characterization
[ edit ]
[ edit ]
Histogram and log-histogram of random deviates from the reciprocal distribution
A positive random variable X is log-uniformly distributed if the logarithm of X is uniform distributed,
ln
(
X
)
∼
U
(
ln
(
a
)
,
ln
(
b
)
)
.
{\displaystyle \ln(X )\sim {\mathcal {U}}(\ln(a ),\ln(b )).}
This relationship is true regardless of the base of the logarithmic or exponential function. If
log
a
(
Y
)
{\displaystyle \log _{a}(Y )}
is uniform distributed, then so is
log
b
(
Y
)
{\displaystyle \log _{b}(Y )}
, for any two positive numbers
a
,
b
≠
1
{\displaystyle a,b\neq 1}
. Likewise, if
e
X
{\displaystyle e^{X}}
is log-uniform distributed, then so is
a
X
{\displaystyle a^{X}}
, where
0
<
a
≠
1
{\displaystyle 0<a\neq 1}
.
Applications
[ edit ]
The reciprocal distribution is of considerable importance in numerical analysis , because a computer ’s arithmetic operations transform mantissas with initial arbitrary distributions into the reciprocal distribution as a limiting distribution.[1]
References
[ edit ]
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Reciprocal_distribution&oldid=1179952303 "
C a t e g o r y :
● C o n t i n u o u s d i s t r i b u t i o n s
H i d d e n c a t e g o r i e s :
● A r t i c l e s w i t h s h o r t d e s c r i p t i o n
● S h o r t d e s c r i p t i o n m a t c h e s W i k i d a t a
● T h i s p a g e w a s l a s t e d i t e d o n 1 3 O c t o b e r 2 0 2 3 , a t 1 4 : 3 7 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
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● A b o u t W i k i p e d i a
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