random signals and systems chapter 10 - ntut.edu.twhtwu/courses/fall2009rp/rp... · 2009-09-09 ·...

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RP Chapter 10 Stochastic Processes 1

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

random signals and systems chapter 10 - ntut.edu.twhtwu/courses/fall2009rp/rp... · 2009-09-09 ·...

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