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ECE 434 RANDOM PROCESSES SPRING 2003
PROBLEM SET 1 Due February 5
Please visit the course website soon: courses.ece.uiuc.edu/ece434/
- Review of Basic Probability
Assigned reading: Chapter one, “Getting started,” of the course notes. It would also be good to review some of the final exams from the last six semesters of ECE313. The past ECE homeworks, exams, and their solutions are available on the ECE 313 website.
Problems to be handed in:
- Simple events A register contains 8 random binary digits which are mutually independent. Each digit is a zero or a one with equal probability. (a) Describe an appropriate probability space (Ω, F, P ) corresponding to looking at the contents of the register. (b) Express each of the following four events explicitly as subsets of Ω, and find their probabilities: E1=“No two neighboring digits are the same” E2=“Some cyclic shift of the register contents is equal to 01100110” E3=“The register contains exactly four zeros” E4=“There is a run of at least six consecutive ones” (c) Find P [E 1 |E 3 ] and P [E 2 |E 3 ].
- Frantic search At the end of each day Professor Plum puts her glasses in her drawer with probability .90, leaves them on the table with probability .06, leaves them in her briefcase with probability 0.03, and she actually leaves them at the office with probability 0.01. The next morning she has no recollection of where she left the glasses. She looks for them, but each time she looks in a place the glasses are actually located, she misses finding them with probability 0.1, whether or not she already looked in the same place. (After all, she doesn’t have her glasses on and she is in a hurry.) (a) Given that Professor Plum didn’t find the glasses in her drawer after looking one time, what is the conditional probability the glasses are on the table? (b) Given that she didn’t find the glasses after looking for them in the drawer and on the table once each, what is the conditional probability they are in the briefcase? (c) Given that she failed to find the glasses after looking in the drawer twice, on the table twice, and in the briefcase once, what is the conditional probability she left the glasses at the office?
- Recognizing cumulative distribution functions Which of the following are valid CDF’s? functions? For each that is not valid, state at least one reason why. For each that is valid, find P (X^2 > 5).
F 1 (x) =
{ e−x^2 4 x <^0 1 − e−x
2 4 x^ ≥^0
F 2 (x) =
{ (^0) x < 0 0 .5 + e−x^0 ≤ x < 3 1 x ≥ 3
F 3 (x) =
{ (^0) x ≤ 0 0 .5 + 20 x 0 < x ≤ 10 1 x ≥ 10
- Distribution of the flow capacity of a network A communication network is shown. The link capacities in megabits per second (Mbps) are given by C 1 = C 3 = 5, C 2 = C 5 = 10 and C 4 =8, and are the same in each direction. Information
Source
1 2
3
4
5
Destination
flow from the source to the destination can be split among multiple paths. For example, if all links are working, then the maximum communication rate is 10 Mbps: 5 Mbps can be routed over links 1 and 2, and 5 Mbps can be routed over links 3 and 5. Let Fi be the event that link i fails. Suppose that F 1 , F 2 , F 3 , F 4 and F 5 are independent and P (Fi) = 0.2 for each i. Let X be defined as the maximum rate (in Mbits per second) at which data can be sent from the source node to the destination node. Find the pmf pX.
- Correlation of histogram values Suppose that n fair dice are independently rolled. Let
Xi =
{ 1 if a 1 shows on the ith^ roll 0 else Yi =
{ 1 if a 2 shows on the ith^ roll 0 else
Let X denote the sum of the Xi’s, which is simply the number of 1’s rolled. Let Y denote the sum of the Yi’s, which is simply the number of 2’s rolled. Note that if a histogram is made recording the number of occurrences of each of the six numbers, then X and Y are the heights of the first two entries in the histogram. (a) Find E[X 1 ] and V ar(X 1 ). (b) Find E[X] and V ar(X). (c) Find Cov(Xi, Yj ) if 1 ≤ i, j ≤ n (Hint: Does it make a difference if i = j?) (d) Find Cov(X, Y ) and the correlation coefficient ρ(X, Y ) = Cov(X, Y )/
√ V ar(X)V ar(Y ). (e) Find E[Y |X = x] for any integer x with 0 ≤ x ≤ n. Note that your answer should depend on x and n, but otherwise your answer is deterministic.
- Transformation of a random variable Let X be exponentially distributed with mean λ−^1. Find and carefully sketch the distribution functions for the random variables Y = exp(X) and Z = min(X, 3).
- Working with a joint density Suppose X and Y have joint density function fX,Y (x, y) = c(1 + xy) if 2 ≤ x ≤ 3 and 1 ≤ y ≤ 2, and fX,Y (x, y) = 0 otherwise. (a) Find c. (b) Find fX and fY. (c) Find fX|Y.
- A function of jointly distributed random variables Suppose (U, V ) is uniformly distributed over the square with corners (0,0), (1,0), (1,1), and (0,1), and let X = U V. Find the CDF and pdf of X.
Extra Credit Problems 1 (Not Required) Prof. B. Hajek Same due date as Problem Set 1 Spring 2003
Topic: σ-algebras, random variables and measurable functions. Suggested reading: Wong and Hajek, Stochastic Processes in Engineering Systems (on reserve in the Engineering Library), pp. 1-14.
Prove the seven statements numbered by Roman numerals.
Definition. Let Ω be an arbitrary set. A nonempty collection F of subsets of Ω is defined to be an algebra if: (a) Ac^ ∈ F whenever A ∈ F and (b) A ∪ B ∈ F whenever A, B ∈ F.
I. If F is an algebra then ∅ ∈ F, Ω ∈ F, and the union or intersection of any finite collection of sets in F is in F. Definition. F is called a σ-algebra if F is an algebra such that whenever A 1 , A 2 , ... are each in F , so is the union, ∪Ai.
II. If F is a σ-algebra and B 1 , B 2 ,... are in F , then so is the intersection, ∩Bi.
III. Let U be an arbitrary nonempty set, and suppose that Fu is a σ-algebra of subsets of Ω for each u ∈ U. Then the intersection ∩u∈U Fu is also a σ-algebra.
IV. The collection of all subsets of Ω is a σ-algebra.
V. If Fo is any collection of subsets of Ω then there is a smallest σ-algebra containing Fo (Hint: use III and IV.)
Definitions. B(R) is the smallest σ-algebra of subsets of R which contains all sets of the form (−∞, a]. Sets in B(R) are called Borel sets. A real-valued random variable on a probability space (Ω, F, P ) is a real-valued function X on Ω such that {ω : X(ω) ≤ a} ∈ F for any a ∈ R.
VI. If X is a random variable on (Ω, F, P ) and A ∈ B(R) then {ω : X(ω) ∈ A} ∈ F. (Hint: Fix a random variable X. Let D be the collection of all subsets A of B(R) for which the conclusion is true. It is enough (why?) to show that D contains all sets of the form (−∞, a] and that D is a σ-algebra of subsets of R. You must use the fact that F is a σ-algebra.)
Remark. By VI, P ({ω : X(ω) ∈ A}), or P (X ∈ A) for short, is well defined for A ∈ B(R).
Definition. A function g mapping R to R is called Borel measurable if {x : g(x) ∈ A} ∈ B(R) whenever A ∈ B(R). VII. If X is a real-valued random variable on (Ω, F, P ) and g is a Borel measurable function, then Y defined by Y = g(X) is also a random variable on (Ω, F, P ).
