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title: "The Jacobian transformation"
canonical: "https://modelassist.epixanalytics.com/space/EA/26575402/The%20Jacobian%20transformation"
format: markdown
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<span style="color: #000000">The Jacobian transformation is an algebraic method for determining the probability distribution of a variable </span>*<span style="color: #000000">y</span>*<span style="color: #000000"> that is a function of just one other variable </span>*<span style="color: #000000">x</span>*<span style="color: #000000"> (i.e. </span>*<span style="color: #000000">y</span>*<span style="color: #000000"> is a transformation of </span>*<span style="color: #000000">x</span>*<span style="color: #000000">) when we know the probability distribution for </span>*<span style="color: #000000">x</span>*<span style="color: #000000">.</span>


- 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:


![image](media://cf9872a9-cd8d-4924-a22f-b0ab5fcdbec2)


> Macro (mathblock)


 > Macro (inline-media-image)

> Macro (mathinline)

is known as the *Jacobian*.



***Example***


If x = Uniform(0,c) and y = 1/x:


> Macro (inline-media-image)

> Macro (mathinline)



> Macro (inline-media-image)

> Macro (mathinline)

 so> Macro (inline-media-image)

 > Macro (mathinline)



> Macro (inline-media-image)

> Macro (mathinline)

 so the Jacobian is> Macro (inline-media-image)

 > Macro (mathinline)



 which gives the distribution for y: > Macro (inline-media-image)

> Macro (mathinline)




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