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 ).
g2 (u,v ) 4.7. JOINT PDFS OF FUNCTIONS OF RANDOM VARIABLES 147 The Jacobian of g , which we denote be J , is the matrix-valued function deﬁned by:
J = J (u, v ) = ∂ g1 (u,v )
∂g2 (u,v )
∂u ∂g1 (u,v )
∂g2 (u,v )
∂v . The Jacobian is also called the matrix derivative of g. Just as for functions of one variable, the
function g near a ﬁxed point uo can be approximated by a linear function:
vo ≈g u
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(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
- The Land