# About probability distributions

#### Sandrine Charles

#### 2023-10-24

This page gives some details about the log-logistic and the log-normal probability distributions as proposed for SSD analyses in MOSAIC. These two distributions are the most use by ecotoxicologists in practice. Both are continuous probability distributions.

# Log-normal distribution

The log-normal distribution applies for non-negative random variables whose logarithm is normally distributed. Thus, if the random variable \(X\) is log-normally distributed, then \(Y = \ln(X)\) has a normal distribution. Similarly, if \(Y\) has a normal distribution, then \(X = e^Y\) has a log-normal distribution.

A log-normal distribution is characterised by two parameters, \(\mu \in [-\infty;+\infty]\) and \(\sigma^2 > 0\) whose values are influencing the curve shape. Below, Fig.1 shows the probability density curves for a fixed value of \(\mu = 0\) and increasing \(\sigma\) values, while Fig.2 shows the corresponding cumulative distribution functions for the same parameter values.

Fig.1: Probability density curves of log-normal distributions with mu = 0 and incresing values of sigma: 1 (dark gray curve), 1.5 (orange (orange curve) and 2 (green curve)

Fig.2: Cumulative distribution functions of a log-normal distribution with a mean equal to 0 and incresing values of sigma: 1 (dark gray curve), 1.5 (orange (orange curve) and 2 (green curve)

# Log-logistic distribution

The log-logistic distribution applies for non-negative random variables whose logarithm has a logistic distribution. It is similar in shape to the log-normal distribution but has heavier tails.

A log-logistic distribution is characterised by two parameters: \(\alpha > 0\), corresponding to the median (also called the scale), and \(\beta > 0\) the shape. Below, Fig.3 shows the probability density curves for a fixed value of \(\alpha = 1\) and increasing \(\beta\) values, while Fig.4 shows the corresponding cumulative distribution functions for the same parameter values.

Fig.3: Probability density curves of log-logistic distributions with alpha = 1 and incresing values of beta: 2 (dark gray curve), 4 (orange (orange curve) and 8 (green curve)

Fig.4: Cumulative distribution functions of log-logistic distributions with alpha = 1 and incresing values of beta: 2 (dark gray curve), 4 (orange (orange curve) and 8 (green curve)

# Log-normal versus log-logistic

Fig. 5 below shows the comparison of cumulative distribution functions of the log-normal and the log-logistic distributions, both distributions having the same mean (\(\mu = \frac{1}{2}\)) and the same variance (\(\sigma^2 = \frac{1}{\pi}-\frac{1}{4}\)).

Fig.5: Comparison of cumulative distribution functions of the log-normal (light gray curve) and the log-logistic (dark gray curve) distributions.