This preview shows page 1. Sign up to view the full content.
Unformatted text preview: ormal to the general Suppose W and Z are independent, standard normal random variables. Their joint pdf is the
product of their individual pdfs: 2
2
α2 +β 2
−α
−β
2
2
e
e
e− 2
fW,Z (α, β ) = √ √ =
.
2π
2π
2π
This joint pdf is called the standard bivariate normal pdf, and it is the special case of the general
bivariate normal pdf obtained by setting the means to zero, the standard deviations to one, and
the correlation coeﬃcient to zero. The general bivariate normal pdf can be obtained from fW,Z by
a linear transformation. Speciﬁcally, if X and Y are related to W and Z by
X
Y
where A is the matrix A= =A W
Z 2
σX (1+ρ)
2 2
σY (1+ρ)
2 + − µX
,
µY
2
σX (1−ρ)
2 2
σY (1−ρ)
2 , then fX,Y is given by (4.37), as can be shown by the method of Section 4.7.1.
Figure 4.26 illustrates the geometry of the bivariate normal pdf. The graph of the standard
bivariate normal, fW,Z , has a bell shape that is spherically symmetric about the origin. The level
sets are thus circles centered at the origin. The peak value of fW,Z is...
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 Tech.
 Spring '08
 Zahrn
 The Land

Click to edit the document details