Editing Probability density function (section)

===Scalar to scalar===

Let <math> g: \Reals \to \Reals</math> be a [[monotonic function]], then the resulting density function is<ref>{{cite web |last1=Siegrist |first1=Kyle |title=Transformations of Random Variables |date=5 May 2020 |url=https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/03%3A_Distributions/3.07%3A_Transformations_of_Random_Variables#The_Change_of_Variables_Formula |publisher=LibreTexts Statistics |access-date=22 December 2023}}</ref>
<math display="block">f_Y(y) = f_X\big(g^{-1}(y)\big) \left| \frac{d}{dy} \big(g^{-1}(y)\big) \right|.</math>

Here {{math|''g''<sup>−1</sup>}} denotes the [[inverse function]].

This follows from the fact that the probability contained in a differential area must be invariant under change of variables. That is,
<math display="block">\left| f_Y(y)\, dy \right| = \left| f_X(x)\, dx \right|,</math>
or
<math display="block">f_Y(y) = \left| \frac{dx}{dy} \right| f_X(x)
= \left| \frac{d}{dy} (x) \right| f_X(x)
= \left| \frac{d}{dy} \big(g^{-1}(y)\big) \right| f_X\big(g^{-1}(y)\big)
= {\left|\left(g^{-1}\right)'(y)\right|} \cdot f_X\big(g^{-1}(y)\big) .</math>

For functions that are not monotonic, the probability density function for {{mvar|y}} is
<math display="block">\sum_{k=1}^{n(y)} \left| \frac{d}{dy} g^{-1}_{k}(y) \right| \cdot f_X\big(g^{-1}_{k}(y)\big),</math>
where {{math|''n''(''y'')}} is the number of solutions in {{mvar|x}} for the equation <math>g(x) = y</math>, and <math>g_k^{-1}(y)</math> are these solutions.