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Adhikari-Sarkar distribution (type 1) Notation
A
d
d
i
t
i
v
e
B
i
n
o
m
i
a
l
(
a
,
n
,
p
)
{\displaystyle \mathrm {AdditiveBinomial} (a,n,p)}
Parameters
n ∈
Z
−
{
0
}
{\displaystyle \mathbb {Z} -\{0\}}
Support
x ∈ {1, 2, 3, ... , 9} PMF
{\displaystyle }
CDF
{\displaystyle }
Mean
{\displaystyle }
Median
{\displaystyle }
Mode
{\displaystyle }
Variance
{\displaystyle }
Skewness
{\displaystyle }
Excess kurtosis
{\displaystyle }
Entropy
{\displaystyle }
MGF
{\displaystyle }
CF
{\displaystyle }
PGF
{\displaystyle }
Also known as the distribution of most significant digit , the probability mass function of the Adhikari-Sarkar distribution (type 1) is given by
A
d
h
i
k
a
r
i
S
a
r
k
a
r
1
(
x
;
n
)
≡
Pr
(
X
=
x
)
=
(
x
+
1
)
1
|
n
|
−
x
1
|
n
|
10
1
|
n
|
−
1
{\displaystyle \mathrm {AdhikariSarkar1} (x;n)\equiv \Pr(X=x)={\frac {(x+1)^{\frac {1}{\left|n\right|}}-x^{\frac {1}{\left|n\right|}}}{10^{\frac {1}{\left|n\right|}}-1}}}
x
=
1
,
2
,
…
,
9
{\displaystyle x=1,2,\dots ,9}
n
∈
Z
−
{
0
}
{\displaystyle n\in \mathbb {Z} -\{0\}}
Cumulative Distribution Function
F
X
(
x
)
=
(
x
+
1
)
1
|
n
|
−
1
10
1
|
n
|
−
1
{\displaystyle F_{X}(x)={\frac {(x+1)^{\frac {1}{\left|n\right|}}-1}{10^{\frac {1}{\left|n\right|}}-1}}}
Expected Value
E
(
X
)
=
−
2
1
|
n
|
+
3
1
|
n
|
+
4
1
|
n
|
+
5
1
|
n
|
+
6
1
|
n
|
+
7
1
|
n
|
+
8
1
|
n
|
+
9
1
|
n
|
−
9
10
1
|
n
|
+
1
10
1
|
n
|
−
1
{\displaystyle \mathbb {E} (X)=-{\frac {2^{\frac {1}{\left|n\right|}}+3^{\frac {1}{\left|n\right|}}+4^{\frac {1}{\left|n\right|}}+5^{\frac {1}{\left|n\right|}}+6^{\frac {1}{\left|n\right|}}+7^{\frac {1}{\left|n\right|}}+8^{\frac {1}{\left|n\right|}}+9^{\frac {1}{\left|n\right|}}-9\ 10^{\frac {1}{\left|n\right|}}+1}{10^{\frac {1}{\left|n\right|}}-1}}}
Symbol
Meaning
∼
{\displaystyle \sim }
X
∼
Y
{\displaystyle X\sim Y}
≡
{\displaystyle \equiv }
the distribution in the title is identical with this distribution
⇐
{\displaystyle \Leftarrow }
the distribution in title is a special case of this distribution
⇒
{\displaystyle \Rightarrow }
this distribution is a special case of the distribution in the title
←
{\displaystyle \leftarrow }
this distribution converges to the distribution in the title
→
{\displaystyle \rightarrow }
the distribution in the title converges to this distribution
Relationship
Distribution
When
⇐
{\displaystyle \Leftarrow }
Furlan's spectrum of power numbers
(
k
,
r
,
R
,
w
)
{\displaystyle \left(k,r,R,w\right)}
k
=
9
r
=
|
n
|
R
=
1
w
=
0
{\displaystyle k=9\qquad r=\left|n\right|\qquad R=1\qquad w=0}
⇐
{\displaystyle \Leftarrow }
Uppuluri-Patil (type 1)
(
j
,
m
,
n
′
)
{\displaystyle \left(j,m,n'\right)}
j
=
1
n
′
=
1
m
=
1
|
n
|
{\displaystyle j=1\qquad n'=1\qquad m={\frac {1}{\left|n\right|}}}
⇐
{\displaystyle \Leftarrow }
Uppuluri-Patil (type 2)
(
j
,
m
,
n
′
)
{\displaystyle \left(j,m,n'\right)}
j
=
1
n
′
=
1
m
=
1
|
n
|
{\displaystyle j=1\qquad n'=1\qquad m={\frac {1}{\left|n\right|}}}
⇐
{\displaystyle \Leftarrow }
Uppuluri-Patil (type 3)
(
j
,
m
)
{\displaystyle \left(j,m\right)}
j
=
1
m
=
1
|
n
|
{\displaystyle j=1\qquad m={\frac {1}{\left|n\right|}}}
→
{\displaystyle \rightarrow }
Benford
n
→
∞
{\displaystyle n\rightarrow \infty }
n
→
−
∞
{\displaystyle n\rightarrow -\infty }
Adhikari, I.S., Sarkar, B.P. (1968). Distribution of most significant digit in certain functions whose arguments are random variables. Sankhyā B 30, 47-58 Uppuluri, V.R.R, Patil, S.A. (1985). The distribution of the first j digits of beta related random variables Communications in Statistics - Simulation and Computation 14, 467-472. Wimmer, G., Altmann. (1999). Thesaurus of univariate discrete probability distributions. Stamm; 1. ed (1999) , pg 3
