Log-normal distribution

The lognormal distribution (short log - normal distribution ) is a continuous probability distribution over the set of positive real numbers. It describes the distribution of a random variable, if it is normally distributed.

• 2.1 maximum
• 2.2 Expectation value
• 2.3 variance
• 2.4 standard deviation
• 2.5 Coefficient of variation
• 2.6 skewness
• 2.7 quantiles
• 2.8 Characteristic Function
• 2.9 moments
• 2:10 moment generating function
• 2:11 entropy

• 3.1 Relationship with the normal distribution

• 4.1 Black- Scholes model
• 4.2 Income Distribution
• 4.3 estimate of sales numbers of companies
• 4.4 Actuarial Mathematics

Definition

Density function

A continuous random variable, subject to the logarithmic normal distribution with the parameter and, when the probability density

Possesses.

Two-dimensional log-normal distribution

Are and two log-normally distributed random variables, then with the transformed correlation coefficient

Their joint probability density defined as

Distribution function

Thus, the log-normal distribution for the distribution function

The distribution function of the lognormal distribution arises in double logarithmically probability paper dar. as a straight line

Properties

Maximum

The probability density takes its maximum value

At the place.

Expected value

Is the expected value of the logarithmic normal distribution

Variance

The variance is given by analogous

Standard deviation

For the standard deviation results

Coefficient of variation

From the expected value and variance are obtained directly the coefficient of variation

Skew

The skewness is given by

That is, the lognormal distribution is skewed to the right.

Quantiles

Is the p- quantile of a normal distribution (ie, the distribution function of the standard normal distribution is ), the p- quantile of the logarithmic normal distribution is given by

In particular, the median, i.e., the value at which the distribution function assumes the value of 0.5, is given by

The greater the difference between the expected value and median, the more pronounced is ia the skewness of a distribution. Here, these parameters differ by a factor. The probability of extremely large expressions is so high at the log-normal distribution.

Characteristic function

The characteristic function is not explicitly represented for the lognormal distribution.

Moments

For the lognormal distribution all moments exist and we have:

Moment generating function

The moment generating function does not exist for the lognormal distribution.

Entropy

The entropy of the lognormal distribution is (expressed in nats )

Relations with other distributions

In the insurance mathematics the distribution of the number of claims is often modeled by means of random variables satisfying the Poisson distribution, the negative binomial distribution - or the logarithmic distribution. On the other hand are suitable for modeling the amount of damages in particular the gamma distribution, the log - gamma distribution or the log - normal distribution.

Relation to the normal distribution

The logarithm of a log- normal random variable is normally distributed. More precisely: If a - distributed random variable (ie, normally distributed with mean and variance ), then the random variable log normally distributed with these parameters and, however, these parameters are not expected value and variance of. Is a certain expected value, and a certain variance is desired, one can easily achieve that by the following formulas:

Applications

Black- Scholes model

In the Black- Scholes model stock prices follow a geometric Brownian motion and are therefore normally distributed logarithmically. This model can be explicitly determine prices of financial options.

Income distribution

Frequently incomes are log-normally distributed. One reason is that there is simply much less bestdotierte positions, the bulk of jobs are more or less low-income, with particularly low incomes are less likely to return. This is exactly the course of most log-normal distributions. This fact can be checked in each operational functioning company.

Estimate of sales numbers of companies

The logarithms of all Fakturenbeträge a company follow approximately a normal distribution. The distance between the logarithm of the logarithm of the smallest and the largest auction invoice represents approximately 6 times the standard deviation of the normal distribution of the logs. This makes it possible, on the mean or expected value of the Fakturenbeträge (see above) to close the log-normal distribution. Multiplying this average by the number of valid invoices results in most cases an acceptable estimate of the magnitude of the turnover of a company; in terms of value it tends to be too high: As often should apply to such estimates and the Benford's law, the Benford distribution should be consulted in these cases. It should be noted that the orders of magnitude ( points ) values ​​of the invoiced amounts are not uniformly distributed, but are approximately normally distributed.

Insurance Mathematics

The lognormal distribution is often used because of the above discussed skewness and the associated major loss slope in the modeling of risks as distribution of the extent of damage. Is the expectation value E and standard deviation specified stdev, one obtains the parameters of the logarithmic normal distribution as follows:

And