Pierre Baldi
   

School of  Information and Computer Sciences (ICS)
Institute for Genomics and Bioinformatics (IGB)
University of California at Irvine ( UCI)
   
 

  


Introduction to Probability and Statistics

ICS /Math 67
Course Reference
Pierre Baldi


Grades

Homeworks

Due by January 15:

p.  53: 1, 3, 5, 10, 17, 23, 24, 36, 46, 55

Due by January 22:

p. 104: 1,  5, 13, 16, 21, 24, 33, 40, 49, 50, 53, 56, 65, 69, 85

Due by January 29:

p. 171: 2, 13, 20, 27,  30, 41, 43, 48, 51, 52, 54, 58, 61, 70

Due by February 5:

p. 228: 1 , 4, 6, 7, 8, 11, 14, 15, 16, 17,18, 19, 20, 21, 23, 24, 26, 27, 28, 29.

Due by February 19:

p. 228: 32,3 4, 39, 40

A coin is flipped 1,000 time s and  700 heads are observed. Using a simple family of models parameterized by p (the probability of heads on a single toss) derive the Maximum Likelihood estimate of p. Using a Beta prior on p, with parameters a and b, derive the Maximum A Posteriori Estimate of p. What happens if the prior is uniform? What if a=b=5? In both cases, plot the prior and posterior densities.

 Due by February 26:

You are interested in estimating the number of email messages received by the UCI campus in one day (email addresses ending in "uci.edu"). Prior to seeing any data, provide a reasonable guess and standard deviation for your estimate. Imagine that on two different days, the actual number of messages is: 210,000  and 220,000. How would this change your estimate and why? Build a Maximum Likelihood and Maximum A Posteriori framework for your answers.

Due by March 5:

p. 471: 2, 3

Write and run a computer program to estimate the area between a normal density function and the x-axis, from minus one to plus one standard deviation from the mean, by simulating a sequence of Bernoulli trials.

p. 290: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,          

Due by March 12:

1. Let X and Y be two  0-1 variables with P(0,0)=0.1, P(0,1)= 0.1, P(1,0)=0.2, P(1,1)=0.6. Computer the marginal  probabilities of X and Y. Are X and Y independent? Computer the  Covariance and the Correlation Coefficient between X and Y.

2. A webmaster of a particular web site is interested in studying the relationship between the indegree X of a web page and the number Y of its visitors per day. After monitoring the web site for a few days he finds the following average values:

X=2 Y= 21; X=3 Y=33; X=5 Y= 55, X=10 Y=99, X=20 Y= 220

 Estimate the number of daily visitors to a web page with indegree: (a) 15; (b) 30.  Explain.

                                                                                                                                                                                          

 Solution to Midterm