Editing Probability density function (section)

===Vector to scalar===

Let <math> V: \R^n \to \R </math> be a differentiable function and <math> X </math> be a random vector taking values in <math> \R^n </math>, <math> f_X </math> be the probability density function of <math> X </math> and <math> \delta(\cdot) </math>  be the [[Dirac delta]] function. It is possible to use the formulas above to determine <math> f_Y </math>, the probability density function of <math> Y = V(X) </math>, which will be given by
<math display="block">f_Y(y) = \int_{\R^n} f_{X}(\mathbf{x}) \delta\big(y - V(\mathbf{x})\big) \,d \mathbf{x}.</math>

This result leads to the [[law of the unconscious statistician]]:
<math display="block">\begin{align}
\operatorname{E}_Y[Y] &=\int_{\R} y f_Y(y) \, dy \\
&= \int_{\R} y \int_{\R^n} f_X(\mathbf{x}) \delta\big(y - V(\mathbf{x})\big) \,d \mathbf{x} \,dy \\
&= \int_{{\mathbb R}^n} \int_{\mathbb R} y f_{X}(\mathbf{x}) \delta\big(y - V(\mathbf{x})\big) \, dy \, d \mathbf{x} \\
&= \int_{\mathbb R^n} V(\mathbf{x}) f_X(\mathbf{x}) \, d \mathbf{x}=\operatorname{E}_X[V(X)].
\end{align}</math>

''Proof:''

Let <math>Z</math> be a collapsed random variable with probability density function <math>p_Z(z) = \delta(z)</math> (i.e., a constant equal to zero). Let the random vector <math>\tilde{X}</math> and the transform <math>H</math> be defined as
<math display="block">H(Z,X)=\begin{bmatrix} Z+V(X)\\ X\end{bmatrix}=\begin{bmatrix} Y\\ \tilde{X}\end{bmatrix}.</math>

It is clear that <math>H</math> is a bijective mapping, and the Jacobian of <math>H^{-1}</math> is given by:
<math display="block">\frac{dH^{-1}(y,\tilde{\mathbf{x}})}{dy\,d\tilde{\mathbf{x}}}=\begin{bmatrix} 1 & -\frac{dV(\tilde{\mathbf{x}})}{d\tilde{\mathbf{x}}}\\ \mathbf{0}_{n\times1} & \mathbf{I}_{n\times n} \end{bmatrix},</math>
which is an upper triangular matrix with ones on the main diagonal, therefore its determinant is 1. Applying the change of variable theorem from the previous section we obtain that
<math display="block">f_{Y,X}(y,x) = f_X(\mathbf{x}) \delta\big(y - V(\mathbf{x})\big),</math>
which if marginalized over <math>x</math> leads to the desired probability density function.