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handout_week7_b - Stochastic Signals and Systems Multiple...

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Stochastic Signals and Systems Multiple Random Variables Virginia Tech Fall 2008 Multiple RVs: Joint CDF The concepts developed so far for handling pairs of random variables are straightforwardly extended to apply to an arbitrary number n of random variables. For any set of n values x 1 , x 2 , . . . , x n the outcomes of the experiment for which X 1 x 1 , X 2 x 2 , . . . , X n < x n constitute an event whose probability F X ( x 1 , x 2 , . . . , x n ) = P [ X 1 x 1 , X 2 x 2 , . . . , X n x n ] where X = { X 1 , X 2 , . . . , X n } , is defined as the joint cumulative distribution function of the random variables X 1 , X 2 , . . . , X n . The cumulative distribution function F X must be such that F X ( -∞ , -∞ , . . . , -∞ ) = 0 F X ( , , . . . , ) = 1
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Multiple RVs: Joint PDF When the cumulative distribution function is a continuous and differentiable function of its variables, we can define the joint probability density function of the n random variables X 1 , X 2 , . . . , X n by f X ( x 1 , x 2 , . . . , x n ) = n x 1 x 2 . . . ∂ x n F X ( x 1 , x 2 , . . . , x n ) It follows directly from these definitions: F X ( x 1 , x 2 , . . . , x n ) = Z x 1 -∞ Z x 2 -∞ . . .
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