1. The Gamma PDF
Eliason (1993)

2. The lognormal PDF
For x > 0, s > 0
So:

3. Lognormal: One tail and no negative values
0.8
0.7
x is always positive
0.6
f(x)
0.5
The lognormal is used for data that reflect the product of many independent variables. For instance, the distribution of the sizes of individuals in a population (DBH) because each individual’s size is the result of many multiplicative processes acting on existing individuals. Continuous, long-tailed, variance greater than the mean.
Applies to data where the outcome is the product of a state and random variable. Log transformed lognormal data are normally distributed. Magnitude of residuals increases with the value of the independent variable. DBH. Sizes of individuals. Growth is a perfect example and this is a very common pattern in plant ecology in general.
Alternative parameter are mumean of log(x) and sigma->variance of log(x).
Can also transform the variables to normal and work with the normal likelihood function instead.
In fact we are often modeling the distribution of errors ---which take both positive or negative values—so, is it better to use a lognormal? What do we use for negative values of errors?
0.4
0.3
0.2
0.1 10 20 30 40 50 60 70 x

4. Standard error
The Standard Error, or Standard Error of the Mean, is an estimate of the standard deviation of the sampling distribution of means, based on the data from one or more random samples.