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University of Illinois at Urbana-Champaign Department of Electrical and Computer Engineering ECE 534: Random Processes Spring 2005 Problem Set 1 Probability Review
Issued: Wednesday, Jan. 26th Due: Beginning of lecture on Monday, Feb. 7
Reading from Hajek: Chapter 1.
Reading from Stark and Woods: Chapters 1, 2 and 3.
Most of this material should be familiar to you from previous courses.
Announcement: The Probability Quiz is scheduled for Wednesday, February 9th, from 5:00- 7:00pm in room DCL1310. The quiz is closed-book; calculators are allowed but will not be necessary.
Problem 1.
From Hajek, Chapter 1: Problems 1, 2, 3, 5, 6, 7 and 9.
Problem 1.
A man has two coins in his pockets: one is a fair coin with a head and a tail, the other is a special coin with two heads. The man picks out a coin randomly, tosses it and observes if he gets a head or a tail. After that, he puts the coin back to the pocket and repeats the procedure for one more time. Specify the set of possible outcomes, i.e., the sample space Ω; the event space, i.e., the σ-field F ; the probability measure P on each event. What is the probability that a head was obtained in the first toss given that the second toss is a tail?
Problem 1.
Consider a probability space (Ω, F, P ).
(a) Under what conditions are A, Ac^ ∈ F independent?
(b) If C ∈ F and P (C) > 0, under what conditions are A, Ac^ conditionally independent given C?
Problem 1.
For a probability space (Ω, F, P ), prove that for all B ∈ F , if P (B) > 0, then the conditional probability P (·|B) is a probability measure defined on F.
Problem 1.
Let
P [[a, b)] =
{ b − a for 0 ≤ a < b ≤ 1 / 2 , 2 3 (b
(^2) − a (^2) ) for 1/ 2 ≤ a < b ≤ 1.
What must P [[a, b)] be in the range a ≤ 1 / 2 < b in order for P to satisfy the axioms of a probability measure? Can P be put in the form P [[a, b)] =
∫ (^) b a f^ (x)dx^ for some appropriate f (x)? If so, what is f (x)?
Problem 1.
Which of the following are valid pdf’s for a continuous random variable X? If the pdf is valid, find the expected value of X; if not, explain why.
(a) f (x) = exp(π(x − 1)), −∞ < x < +∞ ;
(b) f (x) =
{ (^) sin(πx) πx ,^ −^1.^5 < x <^1.^5 , 0 , elsewhere ;
(c) f (x) = 12 e−|x|, −∞ < x < +∞.
Problem 1.
Let X, Y and Z be independent and identically distributed (i.i.d.) nonnegative random variables with density f (α) = e−α^ for α ≥ 0.
(a) Find the probability density of X, conditioned on the event that X ≤ 1.
(b) Find the probability density of X, conditioned on the event that X ≤ 1 and X + Y ≤ 1.
(c) Find the joint density of X and Y , conditioned on the event that X + Y ≤ 1 and X + Y + Z ≥ 1.
(d) Show that the probability density of X conditioned on the event X + Y ≤ 1 and X + Y + Z ≥ 1 is the same as the probability density of the random variable min(U 1 , U 2 ) where U 1 and U 2 are two i.i.d. random variables uniformly distributed on [0, 1].
Problem 1.
The joint pdf of two random variables X and Y is given by
fX,Y (x, y) =
{ A(1 − |x − y|), 0 < x < 1 , 0 < y < 1 0 , otherwise.
(c) Prove that every number x ∈ [0, 1) must be in one of the Sq for some rational number q ∈ [0, 1). Let I be the set of rational numbers in [0, 1); combining (b) and (c) we have that {Si}i∈I is a countable sequence of disjoint sets and ∪i∈I Si = [0, 1).
(d) Use the fact that the Borel measure is translation invariant to prove that every so defined Si is not measurable.
