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Problem Set 4 on Probability with Engineering Applications | ECE 313, 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: Fall 2003;

Typology: Assignments

Pre 2010

Uploaded on 03/16/2009

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UNIVERSITY OF ILLINOIS, URBANA-CHAMPAIGN
Department of Electrical and Computer Engineering
ECE 313: Probability with Engineering Applications-Fall 2003
Problem Set 4: Mean, Variance, and Estimation
Issued: September 19, 2003 Due: September 26, 2003
Reading Assignments: Ross, Chapter 3.1-3.4, Chapter 4.1-4.6
Problems NOT to be turned in:
Problems Theoretical Exercises
Chapter 4 34, 35-38, 43, 48 9, 13, 15
Problems to be turned in:
1. Suppose that 10% of the chips produced by a computer hardware manufacturer are defective. If
100 chips are purchased, does the number of defective chips have a binomial distribution? Explain
your answer.
2. Three dice are rolled. By assuming that each of the 63= 216 possible outcomes is equally likely,
compute P(X=i) for i= 1,2, . . . , 8, where Xis the sum of the 3 dice. Conclusion: It is far
easier to compute the expectation and variance of a sum of i.i.d. r.v.s than it is to compute its pmf !
3. In lecture on September 15, we saw the following implication: for any pair of discrete random
variables Xand Y,
X, Y independent ⇒E[g(X)h(Y)] = E[g(X)]E[h(Y)],for all functions h,g :R→R.
Prove the converse.
4. A box contains 5 red and 5 blue marbles. Two marbles are withdrawn randomly. If they are the
same color, then you win $1.10; if they are different colors, then you win -$1.00 (that is, you lose
$1.00.) Calculate
(a) The expected value of the amount you win.
(b) The variance of the amount you win.
(c) The probability of winning at least $3.00 after ten independent tries. Please include a compu-
tation, and also an application of your favorite bound.
Hint: To simplify calculations, try expressing the winnings as X=aY +b, where Yis a
binomial (10, p) r.v., and a,b, and pare some constants.
5. There are Nmultiple-choice questions on a certain examination. A student knows the answer to K
of these and marks the answer sheet accordingly. For the remaining N−Kquestions, the student
guesses uniformly and randomly among the six choices. The examiner determines W, the number
of incorrect answers on the examiner’s sheet. She wishes to estimate K, based on W.
(a) Explain why the number of wrong answers can be modeled as a binomial random variable W.
What are its parameters?
(b) Obtain the ML estimate of Kbased on the observed value of W.
(c) Do you feel that
c
W=N−
b
K(rather than W) should be used to determine the examination
grade?
1
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Download Problem Set 4 on Probability with Engineering Applications | ECE 313 and more Assignments Statistics in PDF only on Docsity!

UNIVERSITY OF ILLINOIS, URBANA-CHAMPAIGN

Department of Electrical and Computer Engineering

ECE 313: Probability with Engineering Applications-Fall 2003

Problem Set 4: Mean, Variance, and Estimation

Issued: September 19, 2003 Due: September 26, 2003

Reading Assignments: Ross, Chapter 3.1-3.4, Chapter 4.1-4.

Problems NOT to be turned in:

Problems Theoretical Exercises Chapter 4 34, 35-38, 43, 48 9, 13, 15

Problems to be turned in:

  1. Suppose that 10% of the chips produced by a computer hardware manufacturer are defective. If 100 chips are purchased, does the number of defective chips have a binomial distribution? Explain your answer.
  2. Three dice are rolled. By assuming that each of the 6^3 = 216 possible outcomes is equally likely, compute P (X = i) for i = 1, 2 ,... , 8, where X is the sum of the 3 dice. Conclusion: It is far easier to compute the expectation and variance of a sum of i.i.d. r.v.s than it is to compute its pmf!
  3. In lecture on September 15, we saw the following implication: for any pair of discrete random variables X and Y , X, Y independent ⇒ E[g(X)h(Y )] = E[g(X)]E[h(Y )], for all functions h, g : R → R. Prove the converse.
  4. A box contains 5 red and 5 blue marbles. Two marbles are withdrawn randomly. If they are the same color, then you win $1.10; if they are different colors, then you win -$1.00 (that is, you lose $1.00.) Calculate

(a) The expected value of the amount you win. (b) The variance of the amount you win. (c) The probability of winning at least $3.00 after ten independent tries. Please include a compu- tation, and also an application of your favorite bound. Hint: To simplify calculations, try expressing the winnings as X = aY + b, where Y is a binomial (10, p) r.v., and a, b, and p are some constants.

  1. There are N multiple-choice questions on a certain examination. A student knows the answer to K of these and marks the answer sheet accordingly. For the remaining N − K questions, the student guesses uniformly and randomly among the six choices. The examiner determines W , the number of incorrect answers on the examiner’s sheet. She wishes to estimate K, based on W.

(a) Explain why the number of wrong answers can be modeled as a binomial random variable W. What are its parameters? (b) Obtain the ML estimate of K based on the observed value of W. (c) Do you feel that Ŵ = N − K̂ (rather than W ) should be used to determine the examination grade?

  1. Following the approach described in the September 19 lecture,

(a) Prove the following bound for a given positive constant α:

P (X > c) ≤ eΛ(α)−αc, c ∈ R,

where Λ(α) = ln E[eαX^ ]. (b) Compute Λ(α) when X is binomial (10, 12 ). (Hint: see Problem # 3.) (c) What is the best bound in (a) for the random variable in (b)? That is, the minimum of the right hand side over α ∈ R, for a given c ∈ R.. (d) What is the bound obtained using Chebyshev’s bound? (e) Compute P (X > c) in this case with c = 7, and compare to the bounds obtained in (c) and (d). NB: The inequality obtained in (c) is a version of Chernoff’s bound. This is also known as a Large Deviations estimate.