Home page for accesible maths C Proofs of additional results 3.3 Odd central moments of symmetric distributions 3.5 Formal proof of the CLT

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3.4 The formula for a bivariate change of variable

We wish to show that given a 1-1 transformation from two continuous rvs $X$ and $Y$ to new rvs $U(X,Y)$ and $V(X,Y)$ the new density function is

\displaystyle f_{X,Y}(x,y)=f_{U,V}(u,v)|\det(J)|,

where $J$ is the Jacobian matrix, $\partial(u,v)/\partial(x,y)$ . We start with a simple geometric result.

Unnumbered Figure: First link, Second Link

Proposition 3.4.1.

The area of the filled diamond in the left hand figure is $a_{1}b_{2}-a_{2}b_{1}$ .

Proof.

We use the contruction in the right hand figure, find the total shaded area $B=$ (solid+hatched), subtract off the hatched area, $C$ , and then double the result.

Dividing the total shaded region into two rectangles and a triangle we see that

\displaystyle B=a_{2}b_{2}+(a_{1}-a_{2})b_{1}+\frac{1}{2}(a_{1}-a_{2})(b_{2}-b% _{1})=\frac{1}{2}\left(a_{2}b_{2}+a_{1}b_{1}+a_{1}b_{2}-a_{2}b_{1}\right).

However, the red hatched region is

\displaystyle C=\frac{1}{2}a_{1}b_{1}+\frac{1}{2}a_{2}b_{2},

so the blue half-diamond has an area of

\displaystyle B-C=\frac{1}{2}\left(a_{1}b_{2}-a_{2}b_{1}\right).

Doubling gives the required result. ∎

Relabelling $a_{1}\leftrightarrow a_{2}$ and $b_{1}\leftrightarrow b_{2}$ in the figure would lead to an area of $a_{2}b_{1}-a_{1}b_{2}=-(a_{1}b_{2}-a_{2}b_{1})$ , so the general formula is $|a_{1}b_{2}-a_{2}b_{1}|$ .

We consider the small rectangular region $r_{x,y}$ with corners between $(x,y)$ and $(x+\Delta_{x},y+\Delta_{y})$ . For any $(x^{\prime},y^{\prime})$ in this region, Taylor’s theorem gives

\displaystyle\begin{bmatrix}u(x^{\prime},y^{\prime})\\ v(x^{\prime},y^{\prime})\end{bmatrix}\approx\begin{bmatrix}u(x,y)\\ v(x,y)\end{bmatrix}+\begin{bmatrix}u_{x}(x,y)(x^{\prime}-x)+u_{y}(x,y)(y^{% \prime}-y)\\ v_{x}(x,y)(x^{\prime}-x)+v_{y}(x,y)(y^{\prime}-y)\end{bmatrix},

where $u_{x}(x,y)$ is shorthand for $\left.\frac{\partial u}{\partial x}\right|_{x,y}$ etc. Henceforth we abbreviate the derivatives to $u_{x},u_{y},v_{y}$ and $v_{y}$ with the fact that they are evaluated at $(x,y)$ implicit. Similarly, we write $u$ for $u(x,y)$ and $v$ for $v(x,y)$ . So the above formula is equivalent to

\displaystyle\begin{bmatrix}u(x^{\prime},y^{\prime})\\ v(x^{\prime},y^{\prime})\end{bmatrix}\approx\begin{bmatrix}u\\ v\end{bmatrix}+\begin{bmatrix}u_{x}(x^{\prime}-x)+u_{y}(y^{\prime}-y)\\ v_{x}(x^{\prime}-x)+v_{y}(y^{\prime}-y)\end{bmatrix}=\begin{bmatrix}u\\ v\end{bmatrix}+\begin{bmatrix}u_{x}&u_{y}\\ v_{x}&v_{y}\end{bmatrix}\begin{bmatrix}x^{\prime}-x\\ y^{\prime}-y\end{bmatrix}.

Locally, we have an affine transformation, and the multiplying matrix is the Jacobian matrix.

Our rectangle $r_{x,y}$ will therefore transform to a diamond $d_{u,v}$ with corners at

\displaystyle\begin{bmatrix}u\\ v\end{bmatrix}+\begin{bmatrix}u_{x}&u_{y}\\ v_{x}&v_{y}\end{bmatrix}\begin{bmatrix}0&\Delta_{x}&0&\Delta_{x}\\ 0&0&\Delta_{y}&\Delta_{y}\end{bmatrix}.

Setting $a_{1}=u_{x}\Delta_{x}$ , $b_{1}=v_{x}\Delta_{x}$ , $a_{2}=u_{y}\Delta_{y}$ and $b_{2}=v_{y}\Delta_{y}$ , and applying our proposition, we see that the area of $d_{u,v}$ is $|u_{x}v_{y}-u_{y}v_{x}|\Delta_{x}\Delta_{y}$ .

Now $(x,y)\in r_{x,y}\Leftrightarrow(u,v)\in d_{u,v}$ , so $\operatorname{\mathsf{P}}\left({(x,y)\in r_{x,y}}\right)=\operatorname{\mathsf% {P}}\left({(u,v)\in d_{u,v}}\right)$ . But for small enough $\Delta$ , the density is almost constant over the region and so the probability is the height of the density curve multiplied by the area of the region. Thus

\displaystyle f_{X,Y}(x,y)\Delta_{x}\Delta_{y}\approx f_{U,V}(u,v)|u_{x}v_{y}-% u_{y}v_{x}|\Delta_{x}\Delta_{y},

i.e.

\displaystyle f_{X,Y}(x,y)\approx f_{U,V}(u,v)|u_{x}v_{y}-u_{y}v_{x}|.

Letting $\Delta_{x}\downarrow 0$ and $\Delta_{y}\downarrow 0$ gives the required result.

In higher dimensions, Taylor expansion still gives an affine transformation with the Jacobian as the multiplying matrix. That the ‘volume’ of the resulting ‘diamond’ is the determinant of the Jacobian can be shown using the Jordan canonical form, which you will cover in Math220: Linear Algebra.