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# n11 - AT77.02 Signals Systems and Stochastic Processes...

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AT77.02 Signals, Systems and Stochastic Processes Asian Institute of Technology Handout 11 1 Thu 14 Oct 2010 2.1.5 Joint and Conditional CDFs and PDFs The joint CDF of random variables X and Y is de±ned as F XY ( x, y ) = Pr { X x, Y y } Their joint PDF is de±ned as f ( x, y ) = 2 F ( x, y ) ∂x∂y It follows that Pr { x 1 < X x 2 , y 1 < Y y 2 } = i y 2 y 1 i x 2 x 1 f ( x, y ) dxdy. The PDF for X (or Y ) alone is called a marginal PDF of X (or Y ) and can be found from the joint PDF by integrating over the other random variable, i.e. f X ( x ) = i -∞ f ( x, y ) dy, f Y ( y ) = i -∞ f ( x, y ) dx. X and Y are statistically independent (or in short independent ) if f ( x, y ) = f X ( x ) f Y ( y ) for all pairs ( x, y ) The conditional PDF of Y given X is de±ned as f Y | X ( y | x ) = f ( x, y ) f X ( x ) Note that, if X and Y are independent, then f Y | X ( y | x ) = f Y ( y ). Example 2.2 : Suppose that f ( x, y ) = 1 4 e -| x |-| y | . The marginal PDF of X is f X ( x ) = i -∞ 1 4 e -| x |-| y | dy = 1 4 e -| x | i -∞ e -| y | dy = 1 2 e -| x | i 0 e - y dy = 1 2 e -| x | × - e - y v v 0 b ² =1 = 1 2 e -| x | .

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n11 - AT77.02 Signals Systems and Stochastic Processes...

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