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fact, from Section 4.1, that FX (uo ) = FX,Y (uo , +∞) for any uo . Therefore, if X and Y are jointly
FX (uo ) = FX,Y (uo , +∞)
uo ∞ −∞ −∞ fX,Y (u, v )dv du. = (4.3) Equation (4.3) expresses FX (uo ) as the integral over (−∞, uo ] of a quantity in square brackets, so
by deﬁnition, the quantity in square brackets is the pdf of X :
∞ fX (u) = fX,Y (u, v )dv. (4.4) fX,Y (u, v )du. (4.5) −∞ Similarly, ∞ fY (v ) =
−∞ The pdfs fX and fX are called the marginal pdfs of the joint distribution of X and Y. Since X is
trivially a function of X and Y , the mean of X can be computed directly from the joint pdf by
ufX,Y (u, v )dudv.
E [X ] =
−∞ −∞ 124 CHAPTER 4. JOINTLY DISTRIBUTED RANDOM VARIABLES If the integration over v is performed ﬁrst it yields the deﬁnition of E [X ] in terms of the marginal
pdf, fX :
∞ ∞ ∞ u E [X ] =
−∞ ufX (u)du. fX,Y (u, v )dv du =
−∞ −∞ The conditional pdf of Y given X , denoted by fY |X (v |uo ), is undeﬁned if fX (uo ) = 0. It 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 Institute of Technology.
- Spring '08
- The Land