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# EE255/CPS226 Expected Value and Higher Moments - PowerPoint PPT Presentation

EE255/CPS226 Expected Value and Higher Moments. Dept. of Electrical & Computer engineering Duke University Email: bbm@ee.duke.edu , kst@ee.duke.edu. Expected (Mean, Average) Value. Mean, Variance and higher order moments E ( X ) may also be computed using distribution function.

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### EE255/CPS226Expected Value and Higher Moments

Dept. of Electrical & Computer engineering

Duke University

Email: bbm@ee.duke.edu, kst@ee.duke.edu

• Mean, Variance and higher order moments

• E(X) may also be computed using distribution function

Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

• RV’s X and Y (=Φ(X)). Then,

• Φ(X) = Xk, k=1,2,3,.., E[Xk]: kthmoment

• k=1 Mean; k=2: Variance (Measures degree of randomness)

• Example: Exp(λ)  E[X]= 1/ λ; σ2 = 1/λ2

Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

• If Z=X+Y, then

• E[X+Y] = E[X]+E[Y] (X, Y need not be independent)

• If Z=XY, then

• E[XY] = E[X]E[Y] (if X, Y are mutually independent)

Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

• Var[X+Y]=Var[X]+Var[Y] (If X, Y independent)

• Cov[X,Y] E{[X-E[X]][Y-E[Y]]}

• Cov[X,Y] = 0 and (If X, Y independent)

• Cross Cov[ ] terms may appear if not independent.

• (Cross) Correlation Co-efficient:

Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

• For dealing with complex function of rv’s.

• Use transforms (similar z-transform for pmf)

• If X is a non-negative continuous rv, then,

• If X is a non-negative discrete rv, then,

Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

• Complex no. domain characteristics fn. transform is

• If X is Gaussian N(μ, σ), then,

Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

• If Y=aX+b (translation & scaling),then,

• Uniqueness property

Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

• For the LST:

• For the z-transform case:

• For the characteristic function,

Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

• Read sec. 4.5.1 pp.217-227

Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

• R(t) = P(X > t), X: Life-time of a component

• Expected life time or MTTF is

• In general, kthmoment is,

• Series of components, (each has lifetime Exp(λi)

• Overall life time distribution: Exp( ), and MTTF =

Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Series SystemMTTF (contd.)

• RV Xi : ith comp’s life time (arbitrary distribution)

• Case of least common denominator. To prove above

Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

• Parallel system: life time of ith component is rv Xi

• X = max(X1, X2, ..,Xn)

• If all Xi’s are EXP(λ), then,

• As n increases, MTTF also increases as does the Var.

Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

• A system with 1 component and (n-1) cold spares.

• Life time,

• If all Xi’s same,  Erlang distribution.

• Read secs. 4.6.4 and 4.6.5 on TMR and k-out of-n.

• Sec. 4.7 - Inequalities and Limit theorems

Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University