# Solution a a binomial random variable with parameters

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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 matrix-valued 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...
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## 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.

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