How do we formulate scaled inverse chi squared from InverseChiSquareDistribution[v]?

Questions
1

Hello community,
If we input

TransformedDistribution[1/u, u \[Distributed] ChiSquareDistribution[v]]

we get

InverseChiSquareDistribution[v]

Now, even there is a function of InverseChiSquareDistribution[v,s], where v is degrees of freedom and s is scale, for learning purposes, how do we get the same analogy as the inverse of Chi-Square above by using TransformedDistribution of ChiSquareDistribution[v] to get the output of InverseChiSquareDistribution[v,s]?
I am lost where to apply the scale s in the TransformedDistribution function.
Thanks much

2

One can get a distribution where the PDF matches that of InverseChiSquareDistribution[v, s] but I’m not seeing how to get Mathematica to recognize that distribution as a scaled inverse chisquare distribution.

Here is some code to get that distribution and compare the pdf to an inverse chisquare distribution:

distY = TransformedDistribution[1/u, u \[Distributed] ChiSquareDistribution[v]];
distX = TransformedDistribution[s v y, y \[Distributed] distY]
(* TransformedDistribution[\[FormalX] s v, \[FormalX] \[Distributed]  InverseChiSquareDistribution[v]] *)
FullSimplify[PDF[distX, x], Assumptions -> v > 0 && s > 0]

pdf1.png

PDF[InverseChiSquareDistribution[v, s], x]

pdf1.png

3

Thank you very much, Jim.
In simple terms for the correct transformation, the scale factor is to be multiplied by the degrees of freedom for a “lumped scale” multiplier. My mistake was to only multiply the scale factor with the underlying distribution and gave up :slight_smile: