@article {Leak:1965::405,
title = "The J-shaped Probability Distribution",
journal = "",
parent_itemid = "",
publishercode ="",
year = "1965",
volume = "11",
number = "4",
publication date ="1965-12-01T00:00:00",
pages = "405-409",
itemtype = "ARTICLE",
url = "https://www.ingentaconnect.com/content/saf/fs/1965/00000011/00000004/art00006",
author = "Leak, William B.",
abstract = "Up to now, research foresters have lacked an appropriate probability distribution for tree diameter as measured in an uneven-aged stand or forest. A probability distribution--herein called the J-shaped distribution--that should prove applicable to tree diameter and similar variables is derived and described. This distribution belongs to the exponential family, is sometimes called the negative exponential distribution, and has been applied to certain non-forestry problems. The cumulative distribution function is F(x) = 1 - e-rx and the probability density function is f(x) = re-rx where: e is the base for natural logarithms, r > 0 is the one parameter of the distribution, and x 0 is the independent variable (tree diameter). Expressions are presented for the mean and variance; the conditional distribution, mean, and variance; and the characteristic function. The mean of several variables with a common J-shaped distribution is not distributed the same as the component variables; in other words, the J-shaped distribution is not reproductive. Possibly, this probability distribution will be found applicable to certain other continuous variables commonly used in forestry research.",
}