1 RP Chapter 10 Stochastic Processes 1 Random Signals and Systems Chapter 10 RP Chapter 10 Stochastic Processes 2
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Random Signals and Systems
Chapter 10
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Note: Two kinds of Averages
(1) Ensemble average: E[X(t0)]
(2) Time average: time average of x(t,s0) over a period of time
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,...2,1,0!
)]([)( )(01 01 =−
= −− mem
ttmP ttm
Mλλ
Note: The Poisson random variable, M=N(t1)-N(t0) has the PMF
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Note: the Poisson Process is memoryless. That is, the probability of an arrival
duringany instant is independent of the past history of the process.
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Note: The memoryless property of a Poisson process with rate can be viewed
from the exponential interarrival time as follows:λ
xnn etXxtXp λ−=>+> ]|[
Meaning: If the arrival has not occurred by time t, the additional time until
the arrival, Xn – t, has the distribution as Xn.
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Note; Any arrival of N(t) is independently labeled either type 1 with probability p
or type 2 with probability 1-p. Such decomposition of N(t) is called Bernoulli
decomposition.
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Note: (1) The independent increments property of the Brownian motion process:
W(t+r) = W(t)+ [ W(t+r) - W(t) ]
Where W(t+r) - W(t) is independent of W(t)
(2) The Poisson process also holds the independent increments property.
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Note: The Cov [X(t1),X(t2)] reveals how much the process is likely to change over
the time period between t1 and t2.. A high covariance indicates that the
sample function is unlikely to change. A covariance near zero suggests the
rapid change.
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)()()( )()( 11xfxfxf XtXtX == +τNote: for a stationary process x(t),
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End of Chapter 10