04/19/23 1

EE255/CPS226Expected Value and Higher

Moments

Dept. of Electrical & Computer engineering

Duke UniversityEmail: bbm@ee.duke.edu,

kst@ee.duke.edu

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.

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