ece353: probability and random processes lecture 18...

ECE353: Probability and Random Processes

Lecture 18 - Stochastic Processes

Xiao Fu

School of Electrical Engineering and Computer ScienceOregon State University

E-mail: [email protected]

From RV to Stochastic Process

• Recall that a RV X is a mapping from the sample space to a real number (i.e.,X(s)).

5

ECE353 Probability and Random Processes X. Fu, School of EECS, Oregon State University 1


• A random pair is a mapping to two random variables.



• A random vector is a mapping to a sequence of random variables.



• A stochastic process is a mapping X(t, s) that maps an outcome to an infinite-length “sequence” that is indexed by time.


Sample Path

• Sample Path


Sample Path

• Fixing time t = t1, X(t1, s) is a single RV.


Examples

• Example 1: pick up a video on YouTube at random to play.

– Every video is a unique stream of bits.

• Example 2: random sinusoid X(t, s) = A(s) sin(Ω(s)t+ φ(s)).

– “modulation” in communications.

-5 0 5 10

t

-5

0

5X

(t,s

)

-5 0 5 10

t

-2

0

2

X(t

,s)


Types of Stochastic Processes

• Continuous time process: t is continuous. X(t, s).

• Discrete-time process: t is not continuous (e.g., digital signal processing). Xn(s).

• Discrete-valued process: X(t, s) is a discrete RV.

• Continuous-valued process: X(t, s) is continuous RV.

• Q: what is the type of the following:

-5 0 5 10

t

-5

0

5

X(t

,s)

-5 0 5 10

t

-2

0

2

X(t

,s)


Types of Stochastic Processes

• Continuous time process: t is continuous. X(t, s).

• Discrete-time process: t is not continuous (e.g., digital signal processing). Xn(s).

• Discrete-valued process: X(t, s) is a discrete RV.

• Continuous-valued process: X(t, s) is continuous RV.

• Q: what about this:

-5

0

5

X(t

,s)

-5 0 5 10

t

-2

0

2

X(t

,s)

-5 0 5 10

t


Poisson Processes of Rate λ

• Motivation: we wish to model the number of data packages arriving at a datacenter over time; or the number of customer arriving at a mall over time.

• Definition: Poisson Process of Rate λ, denoted by N(t, s) (abused notation N(t)since we know that s is always playing role).

1. N(t) = 0, ∀t < 0;2. for all t > t0, the increment N(t1)−N(t0) is a Poisson RV with mean λ(t1−t0).3. if [t0, t1] and [t′0, t

′1] are non-overlapping, then, the corresponding increments,

N(t1)−N(t0), N(t′1)−N(t′0)

are independent RVs.

• Note: Poisson RV with mean α > 0

PN(n) =

αne

−α

n! , n = 0, 1, 2, . . .

0, o.w.



• Illustration



• Example: Let us assume t1 ≤ t2 ≤ t3 and n1 ≤ n2 ≤ n3. What is the joint PMFPN(t1),N(t2),N(t3)(n1, n2, n3)?



• We are interested in the joint Probability

P [N(t1) = n1, N(t2) = n2, N(t3) = n3]

= P [N(t1)−N(0) = n1, N(t2)−N(t1) = n2 − n1, N(t3)−N(t2) = n3 − n2]

= P [N(t1)−N(0) = n1]P [N(t2)−N(t1) = n2 − n1]P [N(t3)−N(t2) = n3 − n2]

=

(λtn11n1!

e−λt1)(

λ(t2 − t1)n2−n1(n2 − n1)!

e−λ(t2−t1))(

λ(t3 − t2)n3−n2(n3 − n2)!

e−λ(t3−t2))


Arrival Time

• From the Poisson process, one can also have characteristics of the arrival times.

• Let N(t) denote the number of customers that one observe at time t, which isa Poisson process. The time that the first customer arrives is a random variable.The inter-arrival time is also random.


Arrival Time

• Let us consider the arrival time of the first customer, X1. What is the PDF?

• We start with the CDF and P [X1 ≤ x1]. This is not easy to compute, but we maycompute P [X1 > x1] = P [no arrival until time point x1] (note that the numberof arrivals between t = 0 and t = x1 is a Poisson RV with mean λ(x1 − 0)):

P [X1 > x1] = P [N(x1)−N(0) = 0] =λx010!e−λx1 = e−λx1

Hence, FX1(x1) = P [X1 ≤ x1] = 1− e−λx1.

• The PDF is fX1(x1) =dFX1

(x1)

x1= λe−λx1 for x1 ≥ 0:

fX1(x1) =

λe−λx1, x1 ≥ 0

0, o.w.

Beautiful!


Inter-arrival Time• What is the PDF X2, the first inter-arrival time?

• What we know is that X1 = x1 has already happened; and N(x1) = 1.

P [X2 > x2|X1 = x1] = P [N(x1 + x2)−N(x1) = 0|N(x1) = 1]

= P [N(x1 + x2)−N(x1) = 0|N(x1)−N(0) = 1]

= P [N(x1 + x2)−N(x1) = 0] = e−λx2

• The above has nothing to do with x1 ⇒ X2 and X1 are independent; and

FX2(x2) =

1− e−λx2, x2 > 0

0 o.w.

x2 is also an exponentially distributed RV!

• For Poisson N(t) of rate λ, Xi∞i=1: i.i.d. exponential RVs.


Brownian Motion Process

• Definition: The continuous time Brownian Motion: W (t) such that

W (t)|t=0 = W (0) = 0, W (t+ τ)−W (t) ∼ N (0, σ2 = ατ),

i.e., W (t+ τ)−W (t) is a Gaussian RV with variance ατ .

• Discrete-Time Brownian Motion:

Xn+1 = Xn +Wn+1, X0 = 0, Wn ∼ N (0, σ2), Wn∞n=1, i.i.d.

X1 = X0 +W1

X2 = X1 +W2 = W1 +W2

X3 = X2 +W3 = W1 +W2 +W3

...

xn =

∑ni=1Wn, n ≥ 1

0, n ≤ 0.


Brownian Motion Process

• E[Xn] = E[∑ni=1Wi] =

∑ni=1E[Wi] = 0.

• Var[Xn] = Var[∑ni=1Wi] = nσ2 (the variance goes unbounded when n→∞).

• Let Zn = (1/n)Xn = (1/n)∑ni=1Wi.

E[Zn] = 0, Var[Zn] = (1/n)2Var[Xn] =σ2

n.

• The factor 1/n matters so much!


Basic Statistics of Stochastic Process

• Definition: Expected value function of stochastic process X(t) is defined as

µX(t) = E[X(t)].

• Note: µX(t) is a deterministic function that gives the mean of X(t) for all t.

• Discrete-time: µX[n] = E[Xn], for all n ∈ Z.


Basic Statistics of Stochastic Process• Example: Random amplitude cosine process:

X(t) = A cos(ωt+ φ) = A(s)︸︷︷︸random

cos(ωt+ φ).

-5 0 5 10

t

-5

0

5X

(t)

-5 0 5 10

t

-5

0

5

X(t

)

-5 0 5 10

t

-5

0

5

X(t

)



• We can compute

µX(t) = E[X(t)] = E[A cos(ωt+ φ)]

= E[A] cos(ωt+ φ)

• E.g., if A ∼ N (0, 1), then we have

µX(t) = 0, ∀t



• Definition: Auto-covariance of random process:

CX(t, τ) = Cov[X(t), X(t+ τ)]

• Discrete-time:CX[m, k] = Cov[Xm, Xm+k]

• Definition: Auto-correlation of random process:

RX(t, τ) = E[X(t)X(t+ τ)]

RX[m, k] = E[Xm, Xm+k]

CX(t, τ) = RX(t, τ)− µX(t)µX(t+ τ)

CX[m, k] = RX[m, k]− µX[m]µX[m+ k]


Stationary Process• Let us look at a particular time t1: X(t1) is a RV. The PDF fX(t1)(x), generally

speaking, is a function of t.

• Definition: X(t) is stationary if and only if joint PDF

fX(t1),...,X(tm)(x1, . . . , xm) = fX(t1+τ),...,X(tm+τ)(x1, . . . , xm), ∀τ,m

• Hence if X(t) is stationary, fX(t)(x) is the same for all t.


Stationary Process

• Example: Wn∞n=−∞: i.i.d. Gaussian (WGN). Is it stationary?

• How to check? fW1(w1) = fW1+q(w1)? fW1,W2(w1, w2) = fW1+q,W2+q(w1, w2)?

• i.i.d. =⇒ Stationary. (The converse is not true).

• Example: Xn(s) = A(s). Given s, A is fixed (PXn1,Xn2(x1, x2) = P [A2]); alwaysstationary, but not independent.

• Example: Discrete-time Brownian Motion Xn: Var[Xn] = nσ2. Var[X1] = σ2

and Var[X100] = 100σ2. Cannot have the same PDFs.


Stationary Process

• Theorem: If X(t) is a stationary process, then we have

µX(t) = µX, ∀t

andRX(t, τ) = E[X(t)X(t+ τ)] = RX(0, τ) = RX(τ).

• These are necessary conditions of being stationary.

• Proof:

µX(t) = E[X(t)] =

∫ ∞x=−∞

xfX(t)(x)dx

=

∫ ∞x=−∞

xfX(0)(x)dx = µX ∀t,

where we have used stationarity fX(t)(x) = fX(0)(x).


Stationary Process

• For the auto-correlation part:

RX(t, τ) = E[X(t)X(t+ τ)] =

∫ ∞x1=−∞

∫ ∞x2=−∞

x1x2fX(t),X(t+τ)(x1, x2)dx1dx2

=

∫ ∞x1=−∞

∫ ∞x2=−∞

x1x2fX(0),X(τ)(x1, x2)dx1dx2

= RX(0, τ).


Stationary Process

• Necessary conditions are used for disqualifying X(t) as a stationary process.

• Example: Y (t) = A cos(2πfct+ θ); A ∼ N (0, 1) is random. Is Y (t) stationary?

• Sanity check: E[Y (t)] = E[A] cos(2πfct+ θ) = 0.

• Let 2πfct+θ = π/2+2kπ for k ∈ Z. There exist points t′ : πfct′+θ = π/2+2kπ

where Y (s, t′) = 0 for all s. Can this be stationary?

RX(t′, τ) = E[X(t′)X(t′ + τ)] =?


Wide Sense Stationary (WSS) Process

• Definition: X(t) is WSS if and only if

E[X(t)] = µX, ∀t ∈ R

RX(t, τ) = E[X(t)X(t+ τ)] = RX(0, τ), ∀t, ∀τ

• Example: Y (t) = A cos(2πfct+ θ); θ ∼ U [0, 2π] is random. Is Y (t) WSS?

• Let α(t) = 2πfct

E[Y (t)] = AE[cos(α(t) + θ)] = A

∫ 2π

θ=0

cos(α(t) + θ)1

2πdθ

=A

2π

∫ 2π

θ=0

cos(α(t) + θ)dθ = 0.


Wide Sense Stationary (WSS) Process

• In addition, we have

RX(t, τ) = E[X(t)X(t+ τ)] = A2E[cos(2πfct+ θ) cos(2πfc(t+ τ) + θ)]

• Recall that

cosA cosB =1

2cos(A−B) +

1

2cos(A+B).

Hence, we have

RX(t, τ) =A2

2

∫ 2π

θ=0

cos(4πfct+ 2πfcτ + 2θ)dθ

+A2

2

∫ 2π

θ=0

cos(−2πfcτ)dθ

=A2

2

∫ 2π

θ=0

cos(2πfcτ)dθ = RX(0, τ).


ece353: probability and random processes lecture 18...

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