Beta(b, a) beta(beta, alpha), Eta(b, a), Beta – Rockwell Automation Arena Basic Edition Users Guide User Manual
Page 67: Beta(beta, alpha)
A
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S
TATISTICAL
D
ISTRIBUTIONS
61
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A •
Statis
tical
Dis
tribu
tions
Beta(
β
,
α
)BETA(Beta, Alpha)
Shape parameters Beta (
β) and Alpha (α) specified as positive real numbers.
[0, 1] (Can also be transformed to [a,b] as described below)
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.
Probability
Density
Function
x
β 1
–
1 x
–
(
)
α 1
–
B
β α
( , )
---------------------------------------
0 otherwise
where
β
is the complete beta
function given by
B
β α
,
(
)
t
β 1
–
1 t
–
(
)
α 1
–
t
d
0
1
∫
=
f(x) =
for 0 < x < 1
Parameters
Range
Mean
β
β α
+
-------------
Variance
βa
β a
+
(
)
2
β a 1
+ +
(
)
-----------------------------------------------
Applications