Logarithmic distribution
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
owns.
properties
Expected value
The logarithmic distribution has an expected value of
- .
Variance
The variance is determined to be
- .
Coefficient of variation
The coefficient of variation is obtained immediately from the expected value and the variance
- .
Crookedness
The skew results from:
- .
Characteristic function
The characteristic function has the form
- .
Probability generating function
For the probability generating function one obtains
- .
Moment generating function
The moment generating function of the logarithmic distribution is
- .
Iterative computation
The recursive equation applies to the probability function
with start value . This can be used to effectively implement logarithmically distributed random numbers.
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).