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Download a pdf copy of this help file here |
|
Download a pdf copy of this help file here |
The Jacobian transformation is an algebraic method for determining the
probability distribution of a variable y that is a function of
just one other variable x (i.e. y is a transformation of
x) when we know the probability distribution for x.
Let x be a variable with probability density function f(x) and cumulative distribution function F(x);
Let y be another variable with probability density function f(y) and cumulative distribution function F(y);
Let y be related to x by some function such that x and y increase monotonically, then we can equate changes dF(y) and dF(x) together, i.e.:
|f(y)dy| = |f(x)dx|
Rearranging a little, we get:
is known as the Jacobian.
Example
If x = Uniform(0,c) and y = 1/x:
so
so
the Jacobian is 
which gives the distribution for
y: 