The Jacobian transformation


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:

 

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