This preview shows page 1. Sign up to view the full content.
Unformatted text preview: sponds a point (α, β ). This can be written as: α = g1 (u, v )
and β = g2 (u, v ), where g1 and g2 are the two coordinate functions of g , each mapping R2 to R. In
vector notation this can be expressed by α = g1 (u,v) , or, for short, by α = g ( u ).
β
β
v
g2 (u,v ) 4.7. JOINT PDFS OF FUNCTIONS OF RANDOM VARIABLES 147 The Jacobian of g , which we denote be J , is the matrixvalued function deﬁned by:
J = J (u, v ) = ∂ g1 (u,v )
∂u
∂g2 (u,v )
∂u ∂g1 (u,v )
∂v
∂g2 (u,v )
∂v . The Jacobian is also called the matrix derivative of g. Just as for functions of one variable, the
o
function g near a ﬁxed point uo can be approximated by a linear function:
v
g u
v uo
vo ≈g u
uo
−
vo
v +A , where the matrix A is given by A = J (uo , vo ). More relevant, for our purposes, is the related
fact that for a small set R near a point (u, v ), if S is the image of R under the mapping, then
area(S )
area(R) ≈  det(J ).
If W = g ( X ), and we wish to ﬁnd the pdf of W at...
View
Full
Document
This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.
 Spring '08
 Zahrn
 The Land

Click to edit the document details