- Logarithmic distribution
Probability distribution
name =Logarithmic
type =mass
pdf_
cdf_
parameters =0 < p < 1!
support =k in {1,2,3,dots}!
pdf =frac{-1}{ln(1-p)} ; frac{;p^k}{k}!
cdf =1 + frac{Beta_p(k+1,0)}{ln(1-p)}!
mean =frac{-1}{ln(1-p)} ; frac{p}{1-p}!
median =
mode =1
variance =p ;frac{p + ln(1-p)}{(1-p)^2,ln^2(1-p)} !
skewness =
kurtosis =
entropy =
mgf =frac{ln(1 - p,exp(t))}{ln(1-p)}!
char =frac{ln(1 - p,exp(i,t))}{ln(1-p)}!In
probability andstatistics , the logarithmic distribution (also known as the logarithmic series distribution) is adiscrete probability distribution derived from theMaclaurin series expansion:ln(1-p) = p + frac{p^2}{2} + frac{p^3}{3} + cdots.
From this we obtain the identity
: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")-distributedrandom variable ::f(k) = frac{-1}{ln(1-p)} ; frac{p^k}{k}
for k ge 1, and where 0
. Because of the identity above, the distribution is properly normalized.
The
cumulative distribution function is:F(k) = 1 + frac{Beta_p(k+1,0)}{ln(1-p)}
where Beta is the
incomplete beta function .A Poisson mixture of Log("p")-distributed random variables has a
negative binomial distribution . In other words, if N is a random variable with aPoisson distribution , and X_i, i = 1, 2, 3, ... is an infinite sequence of independent identically distributed random variables each having a Log("p") distribution, then:sum_{n=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 applied this distribution to
population genetics .ee also
*
Poisson distribution (also derived from a Maclaurin series)
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