16.322 Stochastic Estimation and Control, Fall 2004 Prof. VanderⅤelde If fw(x, y, = is finite evervwhere, the density fux.(u, v, w)can be found directly by the following steps 1. Evaluate the jacobian of the transformation from X, Y, Z to U,V,w. ou(x,y, =) au(x,y, 2) au(,y, = J(x,y,=) o(x,y2)∂(x,y2)∂(x,y2) aw(x,y,=av(x,y, =)av(x,y,z 2. For every value of u, v, w, solve the transformation equations for x, y, =. If there is more than one solution get all of then i(X, Y, Z x, (u, v, w) f(X,Y,Z)="}→{y(xw) W(X, Y, z)=w 二(l,1. 3. Then fm(u,v,w)= ∑ Jx2(x,y,=) with x;.];, =i given in terms of u,v,w This approach can be applied to the determination of the density function for m variable which are defined to be functions of n variables(n>m) by adding some simple auxiliary variables such as x, y, etc to the list of m so as to total n variables Then apply this procedure and finally integrate out the unwanted auxiliary variables Example: Product u=XY To illustrate this procedure, suppose we are given (x, y) and wish to find the probability density function for the product U=Xr First, define a second random variable; for simplicity, choose V=X. Then use the given 3 step procedure 1. Evaluate the jacobian: (x,y) 2. Solve the transformation equations x=1 Xv=u y=x=7 3. Then find f(u,v)=f(v,16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde If f ( , , ,, xyz , , xyz) is finite everywhere, the density f uvw (,, uvw) can be found directly by the following steps: 1. Evaluate the Jacobian of the transformation from X,Y,Z to U,V,W. ∂uxyz ( , , ) ∂x ∂vxyz ( , , ) J xyz ( , , ) = ∂x ∂wxyz ( , , ) ∂x ∂uxyz ( , , ) ∂y ∂vxyz ( , , ) ∂y ∂wxyz ( , , ) ∂y ∂uxyz ( , , ) ∂z ∂vxyz ( , , ) ∂z ∂wxyz ( , , ) ∂z 2. For every value of u,v,w, solve the transformation equations for x,y,z. If there is more than one solution, get all of them. ( ,, ⎪ i u X Y Z ) = u ⎫ ⎧ x u v w (,, ) v X Y Z ( , , (,, ) = v ⎬ ⎪ → ⎨ y u v w ) i  ( ,, ⎩ w X Y Z ) = w z u v w ) ⎪ ⎭ ⎪ (,, i 3. Then ( , , x y z ) f (,, ,, i i uvw ) = ∑ f xyz i uvw , , J x y z ( , , ) i i i k with xi,yi,zi given in terms of u,v,w. This approach can be applied to the determination of the density function for m variable which are defined to be functions of n variables (n>m) by adding some simple auxiliary variables such as x,y,etc. to the list of m so as to total n variables. Then apply this procedure and finally integrate out the unwanted auxiliary variables. Example: Product U=XY To illustrate this procedure, suppose we are given f x y ( , x y) and wish to find , the probability density function for the product U = XY . First, define a second random variable; for simplicity, choose V = X . Then use the given 3 step procedure. 1. Evaluate the Jacobian: y x Jxy ( , ) = = −x 10 2. Solve the transformation equations: ⎧ x = v xy = u⎫ ⎪ ⎬ ⇒ ⎨ u u v x = ⎭ y = = ⎪ ⎩ x v 3. Then find: 1 f ( , uv ) = f xy (v, u ) uv , , v v Page 3 of 10