EE255/CPS226 Expected Value and Higher Moments Dept. of Electrical & Computer engineering Duke University Email: bbm@ee.duke.edu, kst@ee.duke.edu 7/16/2019

Expected (Mean, Average) Value Mean, Variance and higher order moments E(X) may also be computed using distribution function In case, the summation or the integration does is not absolutely convergent, then E(X) does not exist. 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 shape of the pdf (or pmf) for small and large variance values. σ is commonly referred to as the ‘standard deviation’ Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

E[ ] of mutliple RV’s If Z=X+Y, then If Z=XY, 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, M[θ] is not guaranteed to exist. But for most distributions of our interest, it does exist. 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 Summation in one domain  convolution in the other domain. 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, kth moment is, Series of components, (each has lifetime Exp(λi) Overall life time distribution: Exp( ), and MTTF = The last equality follows from by integrating by parts, int_0^∞ t R’(t) = -t R(t)|0 to ∞ + Int_0^∞ R(t) -t R(t) 0 as t ∞ since R(t)  0 faster than t  ∞. Hence, the first term disappears. Note that the MTTF of a series system is much smaller than the MTTF of an individual component. Failure of any component implies failure of the overall system. 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. These are notes. 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