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EE255/CPS226 Expected Value and Higher Moments. Dept. of Electrical & Computer engineering Duke University Email: [email protected] , [email protected] Expected (Mean, Average) Value. Mean, Variance and higher order moments E ( X ) may also be computed using distribution function.

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ee255 cps226 expected value and higher moments

EE255/CPS226Expected Value and Higher Moments

Dept. of Electrical & Computer engineering

Duke University

Email: [email protected], [email protected]

expected mean average value
Expected (Mean, Average) Value
  • 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

higher moments
Higher Moments
  • 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

e of mutliple rv s
E[ ] of mutliple RV’s
  • 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

variance mutliple rv s
Variance: Mutliple RV’s
  • 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

moment generating function mgf
Moment Generating Function (MGF)
  • 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

mgf contd
MGF (contd.)
  • 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

mgf properties
MGF Properties
  • If Y=aX+b (translation & scaling),then,
  • Uniqueness property

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

mgf properties1
MGF Properties
  • For the LST:
  • For the z-transform case:
  • For the characteristic function,

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

mfg of common distributions
MFG of Common Distributions
  • Read sec. 4.5.1 pp.217-227

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

mttf computation
MTTF Computation
  • 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 system mttf contd
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

mttf computation contd
MTTF Computation (contd.)
  • 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

standby redundancy
Standby Redundancy
  • 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

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