Answer to Question #103368 in Statistics and Probability for Biraj Chhetri

Question #103368
Let X and Y be jointly distributed with p.d.f:


fxy(x,y)=1/4(1+xy) ; |x|>1 |y|>1
=0. ; Otherwise
Show that X and Y are not independent but X2 and Y2 are independent
1
Expert's answer
2020-02-24T11:39:38-0500

Let X and Y be jointly distributed with p.d.f:  


fX,Y(x,y)={14(1+xy)x<1,y<10Otherwise f_{X,Y}(x,y)= \begin{cases} {1 \over 4}(1+xy) & |x|<1, |y|<1 \\ 0 &\text{Otherwise } \end{cases}

Check


fX,Y(x,y)dydx=\displaystyle\int_{-\infin}^{\infin}\displaystyle\int_{-\infin}^{\infin}f_{X,Y}(x,y)dydx=




=111114(1+xy)dydx==\displaystyle\int_{-1}^{1}\displaystyle\int_{-1}^{1} {1 \over 4}(1+xy)dydx=




=1114[(y+xy22)]11dx==\displaystyle\int_{-1}^{1}{1 \over 4}\bigg[(y+{xy^2 \over 2})\bigg]\begin{matrix} 1 \\ -1 \end{matrix}dx=

=1114(1+x(1)22(1+x(1)22))dx==\displaystyle\int_{-1}^{1}{1 \over 4}(1+{x(1)^2 \over 2}-(-1+{x(-1)^2 \over 2}))dx=

=1112dx=12[x]11=\displaystyle\int_{-1}^{1}{1 \over 2}dx={1 \over 2}\big[x\big]\begin{matrix} 1 \\ -1 \end{matrix}

Marginal distribution of X


g(x)=fX,Y(x,y)dy=g(x)=\displaystyle\int_{-\infin}^{\infin}f_{X,Y}(x,y)dy=

=1114(1+xy)dy=14[(y+xy22)]11==\displaystyle\int_{-1}^{1}{1 \over4}(1+xy)dy={1 \over 4}\bigg[(y+{xy^2 \over 2})\bigg]\begin{matrix} 1 \\ -1 \end{matrix}=

=14(1+x(1)22(1+x(1)22))=12={1 \over 4}(1+{x(1)^2 \over 2}-(-1+{x(-1)^2 \over 2}))={1 \over 2}

Marginal distribution of Y


h(y)=fX,Y(x,y)dx=h(y)=\displaystyle\int_{-\infin}^{\infin}f_{X,Y}(x,y)dx=

=1114(1+xy)dx=14[(x+x2y2)]11==\displaystyle\int_{-1}^{1}{1 \over4}(1+xy)dx={1 \over 4}\bigg[(x+{x^2y \over 2})\bigg]\begin{matrix} 1 \\ -1 \end{matrix}=

=14(1+y(1)22(1+y(1)22))=12={1 \over 4}(1+{y(1)^2 \over 2}-(-1+{y(-1)^2 \over 2}))={1 \over 2}

fX(x,y)={12x<1,y<10Otherwise f_{X}(x,y)= \begin{cases} {1 \over 2} & |x|<1, |y|<1 \\ 0 &\text{Otherwise } \end{cases}fY(x,y)={12x<1,y<10Otherwise f_{Y}(x,y)= \begin{cases} {1 \over 2} & |x|<1, |y|<1 \\ 0 &\text{Otherwise } \end{cases}

The support set is a rectangle, so we need to check if it is true that fXY(x,y)=fX(x)fY(y),f_{XY}(x,y)=f_{X}(x)f_{Y}(y), for all (x,y).(x, y). We easily find counterexamples:


fX,Y(12,12)=14(1+12(12))=516f_{X,Y}({1 \over 2},{1 \over 2})={1 \over 4}(1+{1 \over 2}({1 \over 2}))={5 \over 16}

fX(12)fY(12)=12(12)=14f_{X}({1 \over 2})f_{Y}({1 \over 2})={1 \over 2}({1 \over 2})={1 \over 4}

fX,Y(12,12)fX(12)fY(12)f_{X,Y}({1 \over 2},{1 \over 2})\not=f_{X}({1 \over 2})f_{Y}({1 \over 2})

Therefore, X and Y are not independent.


U=X2,V=Y2,0u<1,v<1U=X^2, V=Y^2, 0\leq u<1, \leq v<1

If 1<x<0 and1<y<0-1<x<0 \ and -1<y<0

X=U,Y=VX=-\sqrt{U}, Y=-\sqrt{V}

The Jacobian


J(u,v)=x/ux/vy/uy/v=J(u, v)=\begin{vmatrix} \partial x/\partial u & \partial x/\partial v \\ \partial y/\partial u & \partial y/\partial v \end{vmatrix}==1/(2u)001/(2v)=1/(4uv)=\begin{vmatrix} -1/(2\sqrt{u}) & 0 \\ 0 & -1/(2\sqrt{v}) \end{vmatrix}=1/(4\sqrt{uv})

If 0x<1 and 0y<10\leq x<1 \ and\ 0\leq y<1

X=U,Y=VX=\sqrt{U}, Y=\sqrt{V}


J(u,v)=x/ux/vy/uy/v=J(u, v)=\begin{vmatrix} \partial x/\partial u & \partial x/\partial v \\ \partial y/\partial u & \partial y/\partial v \end{vmatrix}==1/(2u)001/(2v)=1/(4uv)=\begin{vmatrix} 1/(2\sqrt{u}) & 0 \\ 0 & 1/(2\sqrt{v}) \end{vmatrix}=1/(4\sqrt{uv})


14(1+uv)(14uv)=116(1+1uv){1 \over 4}(1+\sqrt{uv})({1 \over 4\sqrt{uv}})= {1 \over 16}(1+{1 \over \sqrt{uv}})

If 1<x<0 and 0y<1-1<x<0 \ and\ 0\leq y<1 or  0x<1 and 0y<1\ 0\leq x<1 \ and\ 0\leq y<1

X=U,Y=VX=-\sqrt{U}, Y=\sqrt{V}

J(u,v)=x/ux/vy/uy/v=J(u, v)=\begin{vmatrix} \partial x/\partial u & \partial x/\partial v \\ \partial y/\partial u & \partial y/\partial v \end{vmatrix}==1/(2u)001/(2v)=1/(4uv)=\begin{vmatrix} -1/(2\sqrt{u}) & 0 \\ 0 & 1/(2\sqrt{v}) \end{vmatrix}=-1/(4\sqrt{uv})

If 0x<1 and 1<y<00\leq x<1 \ and\ -1< y<0

X=U,Y=VX=\sqrt{U}, Y=-\sqrt{V}


J(u,v)=x/ux/vy/uy/v=J(u, v)=\begin{vmatrix} \partial x/\partial u & \partial x/\partial v \\ \partial y/\partial u & \partial y/\partial v \end{vmatrix}==1/(2u)001/(2v)=1/(4uv)=\begin{vmatrix} 1/(2\sqrt{u}) & 0 \\ 0 & -1/(2\sqrt{v}) \end{vmatrix}=-1/(4\sqrt{uv})


14(1uv)(14uv)=116(11uv){1 \over 4}(1-\sqrt{uv})(-{1 \over 4\sqrt{uv}})= {1 \over 16}(1-{1 \over \sqrt{uv}})

If 1<x<0 and1<y<0-1<x<0 \ and -1<y<0


fU,V(u,v)={116(1+1uv)0<u<1,0<v<10Otherwise f_{U,V}(u,v)= \begin{cases} {1 \over 16}(1+{1 \over \sqrt{uv}}) & 0<u<1,0<v<1 \\ 0 &\text{Otherwise } \end{cases}q(u)=fU,V(u,v)dv=q(u)=\displaystyle\int_{-\infin}^{\infin}f_{U,V}(u,v)dv==01116(1+1uv)dv=116(1+1u)=\displaystyle\int_{0}^{1} {1 \over 16}(1+{1 \over \sqrt{uv}})dv={1 \over 16}(1+{1 \over \sqrt{u}})

w(v)=fU,V(u,v)du=w(v)=\displaystyle\int_{-\infin}^{\infin}f_{U,V}(u,v)du==01116(1+1uv)du=116(1+1v)=\displaystyle\int_{0}^{1} {1 \over 16}(1+{1 \over \sqrt{uv}})du={1 \over 16}(1+{1 \over \sqrt{v}})

fU(v)fV(u)=116(1+1u)(116)(1+1v)f_{U}(v)f_{V}(u)={1 \over 16}(1+{1 \over \sqrt{u}})({1 \over 16})(1+{1 \over \sqrt{v}})\not=116(1+1uv)=fU,V(u,v)\not= {1 \over 16}(1+{1 \over \sqrt{uv}})=f_{U,V}(u,v)

Therefore, X2 and Y2 are not independent.


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