MS’25 Assignment 2

Assignment 2

The first part of this course looks at estimators. As such, this assignment will also cover estimators. Necessarily, it will also require things from our first module, namely how to work with functions of random variables.

Again, do not use Mathematica; I want you to immerse yourself in the mathematics to see if you can make connections amongst the parts. This is especially important because there is a theorem in the book that is very useful, but which I skip because I want to encourage you to show you can learn from books. This will be a very important skill for your professional life after Knox.

This assignment also checks that you are paying attention to what does not need to be stated.

Part A: The Geometric Distribution

This section deals with statistics on the Geometric distribution. Let \(Y\) be defined as the number of failures until the first success of independent and identically distributed Bernoulli experiments, each with success probability \(\pi\). We say \( Y \sim GEOM(\pi)\).

Problem 1: The Distributon Function

Prove that the probability mass function for \(Y\) is

\[ f(y;\ \pi) = \left\{\hspace{1em} \begin{array}{ll} \pi (1-\pi)^y & 0 \le y \\[1em] 0 & \text{otherwise} \end{array} \right. \]

Problem 2: Two Moments

Calculate the expected value and the variance of \(Y\).

Problem 3: My Two MoMs

Estimate the method of moments estimator of \(\pi\) based on

the first moment, \(m_1\)
the second moment, \(m_2\)

Feel free to leave your answers in terms of \(m_1\) and \(m_2\).

Problem 4: The MLE

Calculate the maximum likelihood estimator of \(\pi\).

Problem 5: Comparing Estimators

Which of these (two) estimators is best…

in terms of bias?
in terms of mean square error?

To determine this, you may want to use simulation.

Part B: The Log-Uniform Distribution

This section deals with the log-Uniform distribution, which is used in digit tests in forensic accounting. Because of this typical application, \(\theta\) tends to take on values between 2 and 6.

This distribution is defined as the base-10 exponential of a zero-minimum Uniform random variable. In other words, let \(X \sim UNIF(0;\ \theta)\) and define \( Y := 10^X\). Here, \(Y\) has a log-Uniform distribution.

Problem 6: The Distribution Function

Calculate the probability density function of \(Y\).

Problem 7: My One MoM

Estimate the method of moments estimator of \(\theta\) based on the mean, \(m_1\).

Problem 8: The MLE

Calculate the maximum likelihood estimator of \(\theta\). You may want to reread the sections with this problem in mind.

Problem 9: Comparing Estimators

Which of these estimators is best…

in terms of bias?
in terms of mean square error?