- Source: Logarithmic distribution
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
−
ln
(
1
−
p
)
=
p
+
p
2
2
+
p
3
3
+
⋯
.
{\displaystyle -\ln(1-p)=p+{\frac {p^{2}}{2}}+{\frac {p^{3}}{3}}+\cdots .}
From this we obtain the identity
∑
k
=
1
∞
−
1
ln
(
1
−
p
)
p
k
k
=
1.
{\displaystyle \sum _{k=1}^{\infty }{\frac {-1}{\ln(1-p)}}\;{\frac {p^{k}}{k}}=1.}
This leads directly to the probability mass function of a Log(p)-distributed random variable:
f
(
k
)
=
−
1
ln
(
1
−
p
)
p
k
k
{\displaystyle f(k)={\frac {-1}{\ln(1-p)}}\;{\frac {p^{k}}{k}}}
for k ≥ 1, and where 0 < p < 1. Because of the identity above, the distribution is properly normalized.
The cumulative distribution function is
F
(
k
)
=
1
+
B
(
p
;
k
+
1
,
0
)
ln
(
1
−
p
)
{\displaystyle F(k)=1+{\frac {\mathrm {B} (p;k+1,0)}{\ln(1-p)}}}
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
∑
i
=
1
N
X
i
{\displaystyle \sum _{i=1}^{N}X_{i}}
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.
See also
Poisson distribution (also derived from a Maclaurin series)
References
Further reading
Johnson, Norman Lloyd; Kemp, Adrienne W; Kotz, Samuel (2005). "Chapter 7: Logarithmic and Lagrangian distributions". Univariate discrete distributions (3 ed.). John Wiley & Sons. ISBN 978-0-471-27246-5.
Weisstein, Eric W. "Log-Series Distribution". MathWorld.
Kata Kunci Pencarian:
- Logaritma
- Daftar tetapan matematis
- Logarithmic distribution
- Logarithmic
- Log-normal distribution
- Exponential-logarithmic distribution
- Logarithmic scale
- Benford's law
- Beta distribution
- Gamma distribution
- List of probability distributions
- Index of logarithm articles
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