EE255/CPS226 Expected Value and Higher Moments

1
EE255/CPS226Expected Value and Higher Moments

• Dept. of Electrical Computer engineering
• Duke University
• Email bbm_at_ee.duke.edu, kst_at_ee.duke.edu

2
Expected (Mean, Average) Value

• Mean, Variance and higher order moments
• E(X) may also be computed using distribution
  function

3
Higher Moments

• RVs X and Y (F(X)). Then,
• F(X) Xk, k1,2,3,.., EXk kth moment
• k1? Mean k2 Variance (Measures degree of
  randomness)
• Example Exp(?) ? EX 1/ ? s2 1/?2

4
E of mutliple RVs

• If ZXY, then
• EXY EXEY (X, Y need not be independent)
• If ZXY, then
• EXY EXEY (if X, Y are mutually
  independent)

5
Variance Mutliple RVs

• VarXYVarXVarY (If X, Y independent)
• CovX,Y EX-EXY-EY
• CovX,Y 0 and (If X, Y independent)
• Cross Cov terms may appear if not independent.
• (Cross) Correlation Co-efficient

6
Moment Generating Function (MGF)

• For dealing with complex function of rvs.
• 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,

7
MGF (contd.)

• Complex no. domain characteristics fn. transform
  is
• If X is Gaussian N(µ, s), then,

8
MGF Properties

• If YaXb (translation scaling), then,
• Uniqueness property

9
MGF Properties

• For the LST
• For the z-transform case
• For the characteristic function,

10
MFG of Common Distributions

• Read sec. 4.5.1 pp.217-227

11
MTTF Computation

• R(t) P(X gt 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(?i)
• Overall life time distribution Exp( ),
  and MTTF

12
Series System MTTF (contd.)

• RV Xi ith comps life time (arbitrary
  distribution)
• Case of least common denominator. To prove above

13
MTTF Computation (contd.)

• Parallel system life time of ith component is rv
  Xi
• X max(X1, X2, ..,Xn)
• If all Xis are EXP(?), then,
• As n increases, MTTF also increases as does the
  Var.

14
Standby Redundancy

• A system with 1 component and (n-1) cold spares.
• Life time,
• If all Xis 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