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I'm studying for the probability actuarial exam and I came across a problem involving transformations of random variable and use of the Jacobian determinant to find the density of transformed random variable, and I was confused about the general method of finding these new densities. I know the Jacobian matrix is important in the change of variable theorem, but I am having trouble connecting the vector calculus concepts with probability distributions.
Let X, Y have a joint pdf of f(x,y) = e^(-x-y), x>0, y>0. Find the density of U = e^(-x-y).
Solution: The solution creates a new variable V=X=h(u,v), and let's Y= -lnU-V = k(U,V), and finds a new joint pdf g(u,v)=f(V,-lnu-v)*J[h,k] = 1,where J[h,k] is the Jacobian determinant of h, and k. Then, fu(u) = integral(0,-ln u) g(u,v)dv = -ln U, as 0<V<-ln U.
*I am confused why we let V = X...I figure we can we also let V = Y from symmetry and get the same result, but can we let V = 2X, or V= X^2, or V=r(X),r arbitrary function(would r need special properties such as being injective, or bijective?), when computing the density fu. Do
How can we map (x,y) to (u,v)? From my limited understanding, by creating a new variable V, we are mapping the region of positive R^2 to some other region containing of which (u,v) come from. Can anything be said about these regions? Am I missing some fundamental assumptions? How does this relate to the change of variables theorem and its application in finding fu? I am interested in the vector analysis of this problem. Is it possible that there are several mappings with varying probability spaces of (u,v) and varying g(u,v)'s, and integral(limits of v) g(u,v)dv = fu(u), the one distinct density of U (this is what's is what seems strange and fuzzy to me)?
I apologize if any of my questions are not clear. If this is the case, try to clarify the stuff near *.
Let X, Y have a joint pdf of f(x,y) = e^(-x-y), x>0, y>0. Find the density of U = e^(-x-y).
Solution: The solution creates a new variable V=X=h(u,v), and let's Y= -lnU-V = k(U,V), and finds a new joint pdf g(u,v)=f(V,-lnu-v)*J[h,k] = 1,where J[h,k] is the Jacobian determinant of h, and k. Then, fu(u) = integral(0,-ln u) g(u,v)dv = -ln U, as 0<V<-ln U.
*I am confused why we let V = X...I figure we can we also let V = Y from symmetry and get the same result, but can we let V = 2X, or V= X^2, or V=r(X),r arbitrary function(would r need special properties such as being injective, or bijective?), when computing the density fu. Do
How can we map (x,y) to (u,v)? From my limited understanding, by creating a new variable V, we are mapping the region of positive R^2 to some other region containing of which (u,v) come from. Can anything be said about these regions? Am I missing some fundamental assumptions? How does this relate to the change of variables theorem and its application in finding fu? I am interested in the vector analysis of this problem. Is it possible that there are several mappings with varying probability spaces of (u,v) and varying g(u,v)'s, and integral(limits of v) g(u,v)dv = fu(u), the one distinct density of U (this is what's is what seems strange and fuzzy to me)?
I apologize if any of my questions are not clear. If this is the case, try to clarify the stuff near *.
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