Isye 2027

# So l has the memoryless property in discrete time p l

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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...
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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 Tech.

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