Let X and Y be jointly distributed with p.d.f:
fX,Y(x,y)={41(1+xy)0∣x∣<1,∣y∣<1Otherwise Check
∫−∞∞∫−∞∞fX,Y(x,y)dydx=
=∫−11∫−1141(1+xy)dydx=
=∫−1141[(y+2xy2)]1−1dx=
=∫−1141(1+2x(1)2−(−1+2x(−1)2))dx=
=∫−1121dx=21[x]1−1 Marginal distribution of X
g(x)=∫−∞∞fX,Y(x,y)dy=
=∫−1141(1+xy)dy=41[(y+2xy2)]1−1=
=41(1+2x(1)2−(−1+2x(−1)2))=21 Marginal distribution of Y
h(y)=∫−∞∞fX,Y(x,y)dx=
=∫−1141(1+xy)dx=41[(x+2x2y)]1−1=
=41(1+2y(1)2−(−1+2y(−1)2))=21
fX(x,y)={210∣x∣<1,∣y∣<1Otherwise fY(x,y)={210∣x∣<1,∣y∣<1Otherwise The support set is a rectangle, so we need to check if it is true that fXY(x,y)=fX(x)fY(y), for all (x,y). We easily find counterexamples:
fX,Y(21,21)=41(1+21(21))=165
fX(21)fY(21)=21(21)=41
fX,Y(21,21)=fX(21)fY(21) Therefore, X and Y are not independent.
U=X2,V=Y2,0≤u<1,≤v<1
If −1<x<0 and−1<y<0
X=−U,Y=−V
The Jacobian
J(u,v)=∣∣∂x/∂u∂y/∂u∂x/∂v∂y/∂v∣∣==∣∣−1/(2u)00−1/(2v)∣∣=1/(4uv) If 0≤x<1 and 0≤y<1
X=U,Y=V
J(u,v)=∣∣∂x/∂u∂y/∂u∂x/∂v∂y/∂v∣∣==∣∣1/(2u)001/(2v)∣∣=1/(4uv)
41(1+uv)(4uv1)=161(1+uv1)
If −1<x<0 and 0≤y<1 or 0≤x<1 and 0≤y<1
X=−U,Y=V
J(u,v)=∣∣∂x/∂u∂y/∂u∂x/∂v∂y/∂v∣∣==∣∣−1/(2u)001/(2v)∣∣=−1/(4uv) If 0≤x<1 and −1<y<0
X=U,Y=−V
J(u,v)=∣∣∂x/∂u∂y/∂u∂x/∂v∂y/∂v∣∣==∣∣1/(2u)00−1/(2v)∣∣=−1/(4uv)
41(1−uv)(−4uv1)=161(1−uv1)
If −1<x<0 and−1<y<0
fU,V(u,v)={161(1+uv1)00<u<1,0<v<1Otherwise q(u)=∫−∞∞fU,V(u,v)dv==∫01161(1+uv1)dv=161(1+u1)
w(v)=∫−∞∞fU,V(u,v)du==∫01161(1+uv1)du=161(1+v1)
fU(v)fV(u)=161(1+u1)(161)(1+v1)==161(1+uv1)=fU,V(u,v) Therefore, X2 and Y2 are not independent.
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