(→Problem 1: Gone Fishin') |
|||
| Line 17: | Line 17: | ||
*(a) You have a coin of unknown bias. You flip it 10 times, and get TTHHTHTTHT as the sequence of outcomes. What is the maximum likelihood estimate of the bias (i.e. the probability, <math>p</math>, of heads)? | *(a) You have a coin of unknown bias. You flip it 10 times, and get TTHHTHTTHT as the sequence of outcomes. What is the maximum likelihood estimate of the bias (i.e. the probability, <math>p</math>, of heads)? | ||
*(b) A friend has a coin of unknown bias. He flips it <math>n</math> times, and finds that <math>k</math> of them were heads. However, he neglects to record the exact sequence. What is the max-likelihood estimate for the bias in this case? | *(b) A friend has a coin of unknown bias. He flips it <math>n</math> times, and finds that <math>k</math> of them were heads. However, he neglects to record the exact sequence. What is the max-likelihood estimate for the bias in this case? | ||
| + | |||
| + | [[Hamad Al Shehhi_ECE302Fall2008sanghavi]] | ||
== Problem 4: Votes are In == | == Problem 4: Votes are In == | ||
Revision as of 17:16, 1 November 2008
Contents
Instructions
Homework 8 can be downloaded here on the ECE 302 course website.
Problem 1: Gone Fishin'
On average, it takes 1 hour to catch a fish.
- (a) What is (an upper bound on) the probability that it will take 3 hours?
- (b) Landis only has 2 hours to spend fishing. What is (an upper bound on) the probability he will go home fish-less?
Problem 2: Bounded Variance
- (a) What is the maximum variance possible for a Bernoulli random variable?
- (b) What is the maximum variance possible for a binomial random variable, with parameter $ n = 1000 $?
- (c) If $ X $ is uniform on $ [a,b] $, what is its variance?
Problem 3: "Bias" Estimate
- (a) You have a coin of unknown bias. You flip it 10 times, and get TTHHTHTTHT as the sequence of outcomes. What is the maximum likelihood estimate of the bias (i.e. the probability, $ p $, of heads)?
- (b) A friend has a coin of unknown bias. He flips it $ n $ times, and finds that $ k $ of them were heads. However, he neglects to record the exact sequence. What is the max-likelihood estimate for the bias in this case?
Hamad Al Shehhi_ECE302Fall2008sanghavi
Problem 4: Votes are In
The election is over and the votes have been cast. If all the votes were counted, it would show that McCain has won Tippecanoe county. However, the Tippecanoe election officer is lazy however, and decides he is just going to count 1000 random votes. He does so, and finds 600 for Obama and 400 for McCain. He declares Obama has won Tippecanoe. What is (an upper bound on) such an event occuring? (Assume, as in class, that Tippecanoe has an infinite number of people.)
Hint: Let $ p $ be the true fraction of people that voted for Obama. We know that $ p < 1/2 $. Now find and upper bound on the event as a function of $ p $, and then maximize over $ p $.
