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ECE 413: Problem Set 11 - University of Illinois, Assignments of Statistics

University of Illinois - Urbana-ChampaignStatistics

Material Type: Assignment; Class: Probability with Engrg Applic; Subject: Electrical and Computer Engr; University: University of Illinois - Urbana-Champaign; Term: Spring 2005;

Typology: Assignments

Pre 2010

Uploaded on 03/16/2009

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University of Illinois Spring 2005
ECE 413: Problem Set 11
Due: Wednesday April 13 at the beginning of class.
Reminder: Hour Exam II is on Monday April 11, 7 p.m. in 269 Everitt Lab.
Reading:Reading: Ross, Chapters 5 and 6
Noncredit exercises: Ross, Chapter 5, Problems 10-19; 21, 22, 24, 31-41
This Problem Set contains six problems
1. A signal x(t) = exp(−πt2),−∞ < t < ∞, is the input to an ideal low-pass filter whose
transfer function is H(f) = rect(f/2). Let y(t) denote the output of the filter. Find
the numerical value of y(0). [Hint: X(f) = exp(−πf2),−∞ < f < ∞.]
2. [Read Example 3d (pp. 198-199) in Chapter 5 of Ross first] Let the straight line segment
ACB be a diameter of a circle of unit radius and center C. Consider an arc AD of the
circle where the length Xof the arc (measured clockwise around the circle) is a random
variable uniformly distributed on [0,2π). Now consider the random chord AD whose
length we denote by L.
(a) Find the probability that Lis greater than the side of the equilateral triangle
inscribed in the circle.
(b) Express Las a function of the random variable X, and find the pdf for L.
3. Ross, p. 231: Problem 36, Chapter 5.
4. Let Xdenote the time of the first arrival after t= 0 in a Poisson process with arrival
rate λ.
(a) What is the value of the CDF of Xat time T? that is, what is P{X ≤ T}?
(b) Let Adenote the event that there is exactly one arrival in the interval (0, T ].
What is P(A)?
(c) Is the P(A) that you found for part (b) the same as the value of P{X ≤ T}that
you gave in part (a) ? Explain why the two are the same (or are different, as
appropriate).
(d) For 0 < t < T , what is the conditional probability that {X≤t}given the event
A, that is, given that there was exactly one arrival in (0, T ]?
5. Consider a Poisson process with arrival rate λ.
(a) What is the mean number of arrivals in the interval (0,4]? That is, what is
E[N(0,4]]?
(b) What is P[{N(0,3] = 3}∩{N(2,6] = 0}]?
(c) If we observe that there were 5 arrivals in (0,6], what is the maximum-likelihood
estimate of the arrival rate λ?
(d) Now suppose that λ= ln 2. What is the probability that at least one arrival
occurs in (0, t]?
pf2

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University of Illinois Spring 2005

ECE 413: Problem Set 11

Due: Wednesday April 13 at the beginning of class. Reminder: Hour Exam II is on Monday April 11, 7 p.m. in 269 Everitt Lab. Reading:Reading: Ross, Chapters 5 and 6 Noncredit exercises: Ross, Chapter 5, Problems 10-19; 21, 22, 24, 31-

This Problem Set contains six problems

  1. A signal x(t) = exp(−πt^2 ), −∞ < t < ∞, is the input to an ideal low-pass filter whose transfer function is H(f ) = rect(f /2). Let y(t) denote the output of the filter. Find the numerical value of y(0). [Hint: X(f ) = exp(−πf 2 ), −∞ < f < ∞.]

  2. [Read Example 3d (pp. 198-199) in Chapter 5 of Ross first] Let the straight line segment ACB be a diameter of a circle of unit radius and center C. Consider an arc AD of the circle where the length X of the arc (measured clockwise around the circle) is a random variable uniformly distributed on [0, 2 π). Now consider the random chord AD whose length we denote by L.

(a) Find the probability that L is greater than the side of the equilateral triangle inscribed in the circle. (b) Express L as a function of the random variable X , and find the pdf for L.

  1. Ross, p. 231: Problem 36, Chapter 5.

  2. Let X denote the time of the first arrival after t = 0 in a Poisson process with arrival rate λ.

(a) What is the value of the CDF of X at time T? that is, what is P {X ≤ T }? (b) Let A denote the event that there is exactly one arrival in the interval (0, T ]. What is P (A)? (c) Is the P (A) that you found for part (b) the same as the value of P {X ≤ T } that you gave in part (a)? Explain why the two are the same (or are different, as appropriate). (d) For 0 < t < T , what is the conditional probability that {X ≤ t} given the event A, that is, given that there was exactly one arrival in (0, T ]?

  1. Consider a Poisson process with arrival rate λ.

(a) What is the mean number of arrivals in the interval (0, 4]? That is, what is E[N (0, 4]]? (b) What is P [{N (0, 3] = 3} ∩ {N (2, 6] = 0}]? (c) If we observe that there were 5 arrivals in (0, 6], what is the maximum-likelihood estimate of the arrival rate λ? (d) Now suppose that λ = ln 2. What is the probability that at least one arrival occurs in (0, t]?

  1. If hypothesis H 0 is true, the pdf of X is exponential with parameter 5 while if hypothesis H 1 is true, the pdf of X is exponential with parameter 10.

(a) Sketch the two pdfs. (b) State the maximum-likelihood decision rule in terms of a threshold test on the observed value u of the random variable X instead of a test that involves comparing the likelihood ratio Λ(u) = f 1 (u)/f 0 (u) to 1. (c) What are the probabilities of false-alarm and missed detection for the maximum- likelihood decision rule of part(b)? (d) The Bayesian (minimum probability of error) decision rule compares Λ(u) to π 0 /π 1. Show that this decision rule also can be stated in terms of a threshold test on the observed value u of the random variable X. (e) If π 0 = 1/3, what is the average probability of error of the Bayesian decision rule? (f) What is the average error probability of a decision rule that always decides H 1 is the true hypothesis, regardless of the value taken on by X? (g) Show that if π 0 > 2 /3, the Bayesian decision rule always decides that H 0 is the true hypothesis regardless of the value taken on by X. What is the average probability of error for the maximum-likelihood rule when π 0 > 2 /3?