Let for . The Beta function
determines how the integral of this function over the range
varies with and
|
|
|
Properties:
-
1.
-
2.
-
3.
Thus the Beta function can easily be evaluated using the Gamma
function. In R:
We now show that
|
|
|
Firstly, consider the product:
|
|
|
|
|
|
|
|
|
|
|
|
Now apply the transformation and . The Jacobian of
this is
|
|
|
and the range for is now from to , with from to .
Hence
|
|
|
|
|
|
|
|
|
|
|
|