1. EE255/CPS226 Expected Value and Higher Moments
Dept. of Electrical & Computer engineering
Duke University
7/16/2019

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. MFG of Common Distributions
Read sec pp
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

11. 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

12. 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

13. 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

14. Standby Redundancy A system with 1 component and (n-1) cold spares.
Life time,
If all Xi’s same,  Erlang distribution.
Read secs and on TMR and k-out of-n.
Sec Inequalities and Limit theorems
Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University