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4 20 derived distribution derived distribution it is a

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Unformatted text preview: ) = dx λ e −λ x , 0, ENGG2430C lecture 9 if x ≥ 0; otherwise. 4 / 20 Derived Distribution Derived distribution: It is a PMF or PDF of a function of random variables with known distribution. Example: X and Y are two random variables with PDF: (From MIT opencourse 6.041 slides.) Let g (X , Y ) = Y /X (it is a r.v.), computes its PDF. M. Chen ([email protected]) ENGG2430C lecture 9 5 / 20 Derived Distribution: Motivation Sometime we don’t need to compute the PDF. For example, ˆˆ E [g (X , Y )] = g (x , y )fX ,Y (x , y ) dx dy Similarly, Var(g (X , Y )) = E [g 2 (X , Y )] − E 2 [g (X , Y )] can be computed without knowing the PDF of g (X , Y ). Very often it is useful to know the derived distributions In P2P video streaming, packets of the same video frame are coming from multiple peers. Assume one packet per connected peer. The delay to view a video frame is the maximum of end-to-end delays between you and every connected peers. In a storage system that consists of k hard disks, time to the first system hard-disk failure is the minimum of the times to failure of individual hard-disks M. Chen ([email protected]) ENGG2430C lecture 9 6 / 20 Derived Distribution: Discrete R.V. Case Consider a discrete r.v. X and let Y = g (X ). its PMF is given by pY (y ) = P (Y = y ) = ∑ pX (x ) x :g (x...
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