EE255/CPS226 Expected Value and Higher Moments

EE255/CPS226Expected 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 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]: 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 • 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

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 • 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, 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

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 • 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

EE255/CPS226 Expected Value and Higher Moments