I need help in the following problem, I solved it but my answer is wrong (my final answer is different than that of the reference)
I think the limits of the integration are wrong, but I do not know what they should be!
Given the following joint density $$f_{X_1,X_2,X_3}(x_1,x_2,x_3)=\frac{2}{2e-5}x_1^2x_2e^{x_1.x_2.x_3}$$ $$0<x_1,x_2,x_3<1$$
It is required to get the distribution of $$x_1.x_2.x_3$$ I used the transformation $$z=x_1.x_2.x_3$$ $$u=x_1$$ $$v=x_2$$ Hence the inverse functions are $$x_1=u$$ $$x_2=v$$ $$x_3=\frac{z}{uv}$$ I differentiated the inverse functions and got the Jacobian $$|J|=\frac{1}{uv}$$ Hence $$f_{u,v,z}=\frac{2}{2e-5}ue^z$$ where $$0<u,v<1$$ $$0<z<uv$$ To get the required distribution, the distribution of z, I integrated the above function w.r.t u and v . $$f_z=\int_{v=0}^{v=1}\int_{u=0}^{u=1}f_{u,v,z}dudv=\frac{e^z}{2e-5}$$ $$0<z<1$$ However the final answer in the reference is $$f_z=\frac{1}{2e-5}(1-z)^2e^z$$ $$0<z<1$$