 | ||
Parameters 0
<
p
<
1
{\displaystyle 0 Support k
∈
{
1
,
2
,
3
,
…
}
{\displaystyle k\in \{1,2,3,\dots \}\!} pmf −
1
ln
(
1
−
p
)
p
k
k
{\displaystyle {\frac {-1}{\ln(1-p)}}\;{\frac {\;p^{k}}{k}}\!} CDF 1
+
B
(
p
;
k
+
1
,
0
)
ln
(
1
−
p
)
{\displaystyle 1+{\frac {\mathrm {B} (p;k+1,0)}{\ln(1-p)}}\!} Mean −
1
ln
(
1
−
p
)
p
1
−
p
{\displaystyle {\frac {-1}{\ln(1-p)}}\;{\frac {p}{1-p}}\!} Mode 1
{\displaystyle 1} | ||
In probability and statistics, the logarithmic distribution (also known as the logarithmic series distribution or the log-series distribution) is a discrete probability distribution derived from the Maclaurin series expansion
From this we obtain the identity
This leads directly to the probability mass function of a Log(p)-distributed random variable:
for k ≥ 1, and where 0 < p < 1. Because of the identity above, the distribution is properly normalized.
The cumulative distribution function is
where B is the incomplete beta function.
A Poisson compounded with Log(p)-distributed random variables has a negative binomial distribution. In other words, if N is a random variable with a Poisson distribution, and Xi, i = 1, 2, 3, ... is an infinite sequence of independent identically distributed random variables each having a Log(p) distribution, then
has a negative binomial distribution. In this way, the negative binomial distribution is seen to be a compound Poisson distribution.
R. A. Fisher described the logarithmic distribution in a paper that used it to model relative species abundance.
The probability mass function ƒ of this distribution satisfies the recurrence relation