AP Statistics Curriculum 2007 Johnson SB

General Advance-Placement (AP) Statistics Curriculum - Johnson SB Distribution

Johnson SB Distribution

The Johnson SB distribution is related to the normal distribution. Four parameters are needed$\Gamma$, $$\delta$$, $$\lambda$$, $$\epsilon$$ . It is a continuous distribution defined on bounded range $$\epsilon \leq x \leq \epsilon + \lambda$$, and the distribution can be symmetric or asymmetric.

PDF:
$$f(x) = \tfrac{\delta}{\lambda\sqrt{2\pi} z(1-z)} exp(-\tfrac{1}{2}(\gamma + \delta ln(\tfrac{z}{1-z}))^2)$$, where $$z \equiv \tfrac{x-\zeta}{\lambda}$$

CDF:
$$F(x) = \Phi(\gamma + \delta ln \tfrac{z}{1-z})$$, where $$z = \tfrac{x-\epsilon}{\lambda}$$

Moments:
Moments for this distribution do not have a simple expression.

Applications

$$\cdot$$ Epidemiology: http://www.bvsde.paho.org/bvsacd/cd47/data.pdf

$$\cdot$$ Forrestry: http://cms1.gre.ac.uk/conferences/iufro/FMA/SB_Plot_Minimum1.pdf