Parameters b
>
0
{displaystyle b>0}
scale
n
>
0
{displaystyle n>0}
shape
μ
>
0
{displaystyle mu >0}
location Support x
>
μ
{displaystyle x>mu } PDF b
n
(
x
−
μ
)
−
1
−
n
(
e
b
x
−
μ
−
1
)
Γ
(
n
)
ζ
(
n
)
{displaystyle { rac {b^{n}{(x-mu )}^{-1-n}}{left(e^{ rac {b}{x-mu }}-1
ight)Gamma (n)zeta (n)}}}
Where
Γ
(
n
)
{displaystyle Gamma (n)}
is the Gamma function and
ζ
(
n
)
{displaystyle zeta (n)}
is the Riemann zeta function Mean {
μ
+
b
ζ
(
n
−
1
)
(
n
−
1
)
ζ
(
n
)
if
n
>
2
Indeterminate
otherwise
{displaystyle {egin{cases}mu +{ rac {bzeta (n-1)}{(n-1)zeta (n)}}&{ ext{if}} n>2{ ext{Indeterminate}}&{ ext{otherwise}} end{cases}}} Variance {
b
2
(
−
(
n
−
2
)
ζ
(
n
−
1
)
2
+
(
n
−
1
)
ζ
(
n
−
2
)
ζ
(
n
)
)
(
n
−
2
)
(
n
−
1
)
2
ζ
(
n
)
2
if
n
>
3
Indeterminate
otherwise
{displaystyle {egin{cases}{ rac {b^{2}left(-(n-2){zeta (n-1)}^{2}+(n-1)zeta (n-2)zeta (n)
ight)}{(n-2){(n-1)}^{2}{zeta (n)}^{2}}}&{ ext{if}} n>3{ ext{Indeterminate}}&{ ext{otherwise}} end{cases}}} | ||
In statistics, the Davis distributions are a family of continuous probability distributions. It is named after Harold T. Davis (1892–1974), who in 1941 proposed this distribution to model income sizes. (The Theory of Econometrics and Analysis of Economic Time Series). It is a generalization of the Planck's law of radiation from statistical physics.
Contents
Definition
The probability density function of the Davis distribution is given by
where
Background
In an attempt to derive an expression that would represent not merely the upper tail of the distribution of income, Davis required an appropriate model with the following properties