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STAT 243 Assignment 2 - Mean and Variance Algorithms and Random Number Generation - Prof. , Assignments of Statistics

University of California - BerkeleyStatistics
Prof. P. C. Spector

An assignment for a statistics course (stat 243) from the fall, 2009 semester. Students are required to write programs to calculate the mean and variance using three different methods: desk calculator algorithm, method of provisional means, and centering around the first observation. They are also asked to generate uniform and normal random numbers, and test the accuracy and efficiency of their programs. The assignment includes hints and notes for each problem.

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Uploaded on 09/07/2009

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STAT 243 ASSIGNMENT 2 FALL, 2009
due November 9, 2009
1. Mean and Variance Algorithms
Write a program to calculate the mean and variance using the following three different meth-
ods:
(a) Desk Calculator Algorithm
(b) Method of Provisional Means
(c) Centering around the First Observation
Test your program using numbers of your choice, and verify your answer using a program
such as R, SAS, matlab, or a routine from the lapack libraries.
2. Ill-conditioned data
Construct a sequence of data sets such that the coefficient of variation (std. deviation / mean)
becomes increasingly smaller. Continue your sequence until one or more of the algorithms in
part 1 no longer gives correct answers. At what value of the coefficient of variation do you
begin to see problems? At what value of the coefficient of variation does the calculation break
down completely?
Hint: You can generate the sequence you need by adding a progressively larger constant to
any set of numbers.
3. Uniform Random numbers
Write a function to generate uniform random numbers in the range of 0 to 1. Use a multi-
plicative congruential generator, with modulus 232, and a value of your own choosing for a
(but make sure mod(a,8) is 3 or 5.) Calculate the mean and variance for several sequences
of numbers from your generator, using one of the programs written for part 1. What is the
theoretical value of the mean and variance for a uniform distribution from 0 to 1? How do
your observed values compare to these values?
4. Normal Random Numbers
Write a program to generate normally distributed random numbers with mean 0 and variance
1 by a) using the Polar method, and b) adding together six uniform random numbers. Ex-
amine the means, variances, minimums and maximums from each of ten samples from each
method to insure that they look reasonable. Compare the time it takes to generate normal
random variables by the two methods.
Note: In order to time a program, you can use the system call getrusage(). To make things
easier, look at the file timer.h in ../s243/samples.
EXTRA CREDIT
Using one or more of the methods described in class, test your random number generator or
the random generators from R, matlab, SAS, or some other source.

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Download STAT 243 Assignment 2 - Mean and Variance Algorithms and Random Number Generation - Prof. and more Assignments Statistics in PDF only on Docsity!

STAT 243 ASSIGNMENT 2 FALL, 2009

due November 9, 2009

1. Mean and Variance Algorithms

Write a program to calculate the mean and variance using the following three different meth- ods: (a) Desk Calculator Algorithm (b) Method of Provisional Means (c) Centering around the First Observation Test your program using numbers of your choice, and verify your answer using a program such as R, SAS, matlab, or a routine from the lapack libraries.

2. Ill-conditioned data

Construct a sequence of data sets such that the coefficient of variation (std. deviation / mean) becomes increasingly smaller. Continue your sequence until one or more of the algorithms in part 1 no longer gives correct answers. At what value of the coefficient of variation do you begin to see problems? At what value of the coefficient of variation does the calculation break down completely? Hint: You can generate the sequence you need by adding a progressively larger constant to any set of numbers.

3. Uniform Random numbers

Write a function to generate uniform random numbers in the range of 0 to 1. Use a multi- plicative congruential generator, with modulus 2^32 , and a value of your own choosing for a (but make sure mod(a,8) is 3 or 5.) Calculate the mean and variance for several sequences of numbers from your generator, using one of the programs written for part 1. What is the theoretical value of the mean and variance for a uniform distribution from 0 to 1? How do your observed values compare to these values?

4. Normal Random Numbers

Write a program to generate normally distributed random numbers with mean 0 and variance 1 by a) using the Polar method, and b) adding together six uniform random numbers. Ex- amine the means, variances, minimums and maximums from each of ten samples from each method to insure that they look reasonable. Compare the time it takes to generate normal random variables by the two methods. Note: In order to time a program, you can use the system call getrusage(). To make things easier, look at the file timer.h in ../s243/samples.

EXTRA CREDIT

Using one or more of the methods described in class, test your random number generator or the random generators from R, matlab, SAS, or some other source.