ee255/cps226 expected value and higher moments
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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. - PowerPoint PPT PresentationTRANSCRIPT
04/19/23 1
EE255/CPS226Expected Value and Higher
Moments
Dept. of Electrical & Computer engineering
Duke UniversityEmail: [email protected],
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University
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
RV’s X and Y (=Φ(X)). Then,
Φ(X) = Xk, k=1,2,3,.., E[Xk]: kth moment 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
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
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)
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.)
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
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,
MGF Properties
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University
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
R(t) = P(X > t), X: Life-time of a component Expected life time or MTTF is
In general, kth moment 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.)
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.)
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.