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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 (IE@CUHK) 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 endtoend delays between
you and every connected peers.
In a storage system that consists of k hard disks, time to the ﬁrst
system harddisk failure is the minimum of the times to failure of
individual harddisks
M. Chen (IE@CUHK) 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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 Spring '14

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