Getting Started

A • STATISTICAL DISTRIBUTIONS
Beta(, ): BETA(Beta, Alpha)
Probability
Density
Function
f(x) =
x – 1 1 – x – 1
---------------------------------------
B( , )
0 otherwise
where is the complete beta
function given by
1
B
= t – 1 1 – t – 1 dt
0
for 0 < x < 1
Parameters
Shape parameters Beta () and Alpha () specified as positive real numbers.
Range
[0, 1] (Can also be transformed to [a,b] as described below)
Mean
------------
+
Variance
a
----------------------------------------------
+ a 2 + a + 1
Applications
Because of its ability to take on a wide variety of shapes, this distribution is often
used as a rough model in the absence of data. Also, because the range of the beta
distribution is from 0 to 1, the sample X can be transformed to the scaled beta sample
Y with the range from a to b by using the equation Y = a + (b - a)X. The beta is often
used to represent random proportions, such as the proportion of defective items in a
lot.
153