The conditional density fy x v uo is undened if uo 0

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Unformatted text preview: ws. Start with the fact, from Section 4.1, that FX (uo ) = FX,Y (uo , +∞) for any uo . Therefore, if X and Y are jointly continuous-type, 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 LOTUS: ∞ ∞ 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 defi...
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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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