For defined in the condition of random values:
If the components are XI=(XI1;XI2) are independent, then calculate the conditional mathematical expectation of E(min{XI1; XI2}|max{XI1; XI2}), otherwise find E(XI1|XI2).
The random values of the XI1 and XI2 are evenly distributed in the quadrangle with peaks (2, 2), (1, -2), (-2, -2), (-2, 2).
My solution:
$E(min(\xi_{1};\xi_{2})|max(\xi_{1};\xi_{2}))=E(V|Z);$
$F_\xi(z)=P(Z<z)=P(max(\xi_{1};\xi_{2})<z)=P(\xi_{1}<z)\cdot P(\xi_{2}<z)= F_{\xi_{1}}(z)\cdot F_{\xi_{2}}(z);$
$f_z(z)=f_{\xi_1}(z)\cdot F_{\xi_{2}}(z)+F_{\xi_{1}}(z)\cdot f_{\xi_2}(z);$
$F_{y;z}(y;z)=P(min(\xi_1;\xi_2)<y|max(\xi_1;\xi_2)<z)=P(max(\xi_1;\xi_2)<z)-P(min(\xi_1;\xi_2)>=y,max(\xi_1;\xi_2)<z)=F_{\xi_{1}}(z)\cdot F_{\xi_{2}}(z)-(F_{\xi_{1}}(z)-F_{\xi_{1}}(y))\cdot(F_{\xi_{2}}(z)-F_{\xi_{1}}(y));$
$f_{y;z}(y;z)=\frac{\partial^2}{\partial y\partial z}(F_{y|z}(y;z))=\frac{\partial^2}{\partial y\partial z}(F_{\xi_{1}}(z)\cdot F_{\xi_{2}}(z)-(F_{\xi_{1}}(z)-F_{\xi_{1}}(y))\cdot(F_{\xi_{2}}(z)-F_{\xi_{2}}(y)))=\frac{\partial}{\partial z}(f_{\xi_{1}}(y)\cdot F_{\xi_{2}}(z)+f_{\xi_{1}}(y)\cdot F_{\xi_{2}}(z)-(f_{\xi_{1}}(y)\cdot F_{\xi_{2}}(y)+f_{\xi_{2}}(y)\cdot F_{\xi_{1}}(y)))=f_{\xi_{1}}(z)\cdot f_{\xi_{2}}(y)+f_{\xi_{1}}(y)\cdot f_{\xi_{2}}(z);$
$f_{y|z}(y|z)=\dfrac{f(y;z)^{'}(y;z)}{f_z(z)}=\dfrac{f_{\xi_{1}}(z)\cdot f_{\xi_{2}}(y)+f_{\xi_{1}}(y)\cdot f_{\xi_{2}}(z)}{f_{\xi_{1}}(z)\cdot F_{\xi_{2}}(z)+F_{\xi_{1}}(z)\cdot f_{\xi_{2}}(z)};$
$F_{y|z}(y;z)_1=\dfrac{f_{\xi_{11}}(z)\cdot f_{\xi_{2}}(y)+f_{\xi_{11}}(y)\cdot f_{\xi_{2}}(z)}{f_{\xi_{11}}(z)\cdot F_{\xi_{2}}(z)+F_{\xi_{11}}(z)\cdot f_{\xi_{2}}(z)};$
$F_{y|z}(y;z)_2=\dfrac{f_{\xi_{12}}(z)\cdot f_{\xi_{2}}(y)+f_{\xi_{12}}(y)\cdot f_{\xi_{2}}(z)}{f_{\xi_{12}}(z)\cdot F_{\xi_{2}}(z)+F_{\xi_{12}}(z)\cdot f_{\xi_{2}}(z)};$
$F_{y|z}(y;z)=F_{y|z}(y;z)_1+F_{y|z}(y;z)_2;$
${E_{y|z}}_1=\int^{2}_{-2} y\cdot F_{y|z}(y;z)_1\,dy;$
${E_{y|z}}_2=\int^{2}_{-2} y\cdot F_{y|z}(y;z)_2\,dy;$
$E_{y|z}={E_{y|z}}_1+{E_{y|z}}_2.$
FXi2[y_]=Integrate[fXi2[y],{y,-2,y}]
FXi11[x_]=Integrate[fXi11[x],{x,-2,x}]
FXi12[x_]=Integrate[fXi12[x],{x,4*x-6,x}]
FXi1[x_]=FXi11[x]+FXi12[x]//FullSimplify
Fyz11[y_,z_]=(fXi11[z]*fXi2[y]+fXi11[y]*fXi2[z])*(fXi11[z]*FXi2[z]+FXi11[z]*fXi2[z])//FullSimplify
Fyz12[y_,z_]=(fXi12[z]*fXi2[y]+fXi12[y]*fXi2[z])*(fXi12[z]*FXi2[z]+FXi12[z]*fXi2[z])//FullSimplify
Fyz[y_,z_]=Fyz11[y,z]+Fyz12[y,z]//FullSimplify
Eyz11[z_]=Integrate[y*Fyz11[y,z],{y,-2,2}]//FullSimplify
Eyz12[z_]=Integrate[y*Fyz12[y,z],{y,4*x-6,2}]//FullSimplify
Eyz[z_]=Eyz12[z]+Eyz11[z]//FullSimplify
Calculation results:
photo № 1: https://i.stack.imgur.com/OlgUw.png
photo № 2: https://i.stack.imgur.com/Q0XuA.png
photo № 3: https://i.stack.imgur.com/aeQbp.png
Please check my solution. Tell me where there are errors and how you would correct them.