Logarithmic distribution

Probability function of the logarithmic distribution for (blue), (green) and (red) ${\displaystyle p=0.2}$ ${\displaystyle p=0.5}$ ${\displaystyle p=0.8}$

The logarithmic distribution is a probability distribution in stochastics . It is univariate , a discrete probability distribution and comes from the field of actuarial mathematics . It is interesting as a damage amount distribution, but is rarely used to determine the number of damage.

definition

A discrete random variable satisfies the logarithmic distribution with the parameter (probability of success) if it is the probability ${\displaystyle X\in \mathbb {N} \setminus \left\{0\right\}}$ ${\displaystyle p}$

${\displaystyle \operatorname {P} (X=k)=f(k)=-{\frac {p^{k}}{k}}\cdot {\frac {1}{\ln(1-p)}}}$

owns.

properties

Expected value

The logarithmic distribution has an expected value of

${\displaystyle \operatorname {E} (X)=-{\frac {p}{(1-p)\ln(1-p)}}}$.

Variance

The variance is determined to be

${\displaystyle \operatorname {Var} (X)=-{\frac {p\cdot (\ln(1-p)+p)}{(1-p)^{2}\ln ^{2}(1-p)}}}$.

Coefficient of variation

The coefficient of variation is obtained immediately from the expected value and the variance

${\displaystyle \operatorname {VarK} (X)={\sqrt {-{\frac {p}{\ln(1-p)+p}}}}}$.

Crookedness

The skew results from:

${\displaystyle \operatorname {v} (X)={\frac {1}{\sqrt {p(-\ln(1-p)-p)}}}\left({\frac {\ln ^{2}(1-p)}{-\ln(1-p)-p}}+p(-\ln(1-p)-2p)\right)}$.

Characteristic function

The characteristic function has the form

${\displaystyle \phi _{X}(s)={\frac {\ln(1-pe^{is})}{\ln(1-p)}}}$.

Probability generating function

For the probability generating function one obtains

${\displaystyle g_{X}(s)={\frac {\ln(1-ps)}{\ln(1-p)}}}$.

Moment generating function

The moment generating function of the logarithmic distribution is

${\displaystyle m_{X}(s)={\frac {\ln(1-pe^{s})}{\ln(1-p)}}}$.

Iterative computation

The recursive equation applies to the probability function

${\displaystyle f(k+1)={\frac {kp}{k+1}}f(k)}$

with start value . This can be used to effectively implement logarithmically distributed random numbers. ${\displaystyle f(1)={\frac {-p}{\ln(1-p)}}}$

Relationship to other distributions

If you combine the logarithmic distribution with the composite Poisson distribution , the result is the negative binomial distribution and, as a special case, the geometric distribution .

literature

• Hans-Otto Georgii: Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , doi : 10.1515 / 9783110215274 .

Web links

• Eric W. Weisstein : Logarithmic Distribution . In: MathWorld (English).