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University of Illinois at Urbana-Champaign
ECE 534: RANDOM PROCESSES
Fall 2006
Midterm 1
Monday, October 16, 2006
Name:
- This is a closed-book exam. You may consult both sides of one sheet of notes, typed in font size 10 or equivalent handwriting size.
- Calculators, laptop computers, Palm Pilots, two-way email pagers, etc. may not be used.
- Write your answers in the space provided.
- Please show all of your work. Answers without appropriate justification will receive very little credit.
Score:
- (12 points)
- (12 points) Total. (24 points)
Problem 1. (Note that each part of this problem can be solved indepen- dently by using what has been shown in the previous part.) Let (Ω, F, P ) be a probability space. Define:
An ∈ F, n = 1, 2 ,... Bm =
n≥m
An, m = 1, 2 ,... (1)
a) EXTRA CREDIT (although this doesn’t take much effort to show:) Show that
A {ω s.t. ω ∈ An finitely often } (2)
can be expressed as
A =
⋃^ ∞
m=
n≥m
Acn
c) Let S be a random variable along with the sequence S 1 , S 2 ,... of random variables, all defined on (Ω, F, P ). Define An() = {ω : |Sn(ω) − S(ω)| > } and Bm() along with A() in terms of An() according to (1) and (2). Show that if (^) ∞ ∑
n=
P (An()) < ∞ for all > 0
then Sn →a.s. S Hint: use the result of part b).
d) Consider a random variable X with expectation μ and suppose that for any > 0, l(μ + ) > 0 and l(μ − ) > 0 where
M (θ) E
[
eθX^
]
l(a) max θ θa − log M (θ).
Let X 1 ,... , Xn be a sequence of i.i.d. random variables distributed identically to X. Prove the strong law of large numbers. Hint: use the result of part c).
b) Find the linear MMSE estimator of Z given Y.
c) Are X and W uncorrelated? Are X and W independent?
