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Assignment 4
Welcome to the assignment testing the fourth part of this course: hypothesis testing. This assignment focuses on the Exponential distribution and functions of it.
The Exponential distribution can be parameterized in a couple of ways. For this assignment, let us parameterize it based on its mean, \(\theta\). With this, its probability density function is:
\[
f(x;\ \theta) :=\ \left\{
\begin{array}{ll}
\frac{1}{\theta}\ e^{-x/\theta} \hspace{2em} & x \ge 0 \\ 0 & \text{otherwise}
\end{array}
\right.
\]
The Exponential distribution is used to model lifetimes of simple systems. As such, it is popular with actuaries. Functions of Exponential random variables can be used to model more complicated systems. In this assignment, we will be looking at a couple of ways of estimating the value of \(\theta\), which is the mean time until failure.
Again, do not use Mathematica; I still want you to immerse yourself in the mathematics and look for connections. A good scientist will search for such connections to see a deeper meaning to the mathematics being performed.
Note: Either do 4 or 9, not both. You will not receive extra credit for doing both. Select one or the other.*
Part A: Sums of Exponentials
In this section, let us look at the sum of those Exponential random variables. Specifically, let us have a random sample of size \(n\), where \(X_i \sim \text{Exp}(\theta)\). Let us now define \(T\) as the sample sum.
\[
T := \sum_{i=1}^n\ X_i
\]
Problem 1: The Probability Function
Calculate the probability density function of \(T\).
Problem 2: The MLE for \(\theta\)
Calculate the maximum likelihood estimator for \(\theta\), where \(n\) is known. This estimator needs to be based on your distribution in Problem 1.
Problem 3: A Confidence Interval
Using whatever methods you have available to you, find the endpoints of a central 95% confidence interval for \(\theta\), when \(n = 4\).
Problem 4: Neyman and Pearson*
Finally, use the Neyman–Pearson Lemma to find a value of \(K\) for testing
\[
\begin{align*}
H_0:& \hspace{1ex} \theta = 1 \\
H_A:& \hspace{1ex} \theta = 4
\end{align*}
\]
for \(n = 4\). For consistency, let us keep \(\alpha = 0.05\) here.
Part B: Minimums of Exponentials
Let us now model the minimum of m Exponential random variables. Specifically, let one experiment be observing the minimum time-until-failure for a group of \(m\) Exponentially-distributed random variables. That is, let us have a random sample of size \(m\), where \(X_j \sim \text{Exp}(\theta)\). Next, let us define \(W\) as the minimum of those values:
\[
W := \min\left\{ X_j \right\}
\]
Finally, let us take a sample of size \(n\) from this \(W\) distribution. In other words, let us perform the above experiment \(n\) times and record the value of \(W_i\) each time.
Problem 5: The Probability Function
Calculate the probability density function of \(W\).
Problem 6: Is It Exponential Class?
Determine if \(W\) is a member of the Exponential Class of distributions. [The book refers to this as the Exponential Family.]
Problem 7: Sufficiency
If your distribution is EC, we can easily determine a sufficient statistic for \(\theta\). What is that sufficient statistic for \(\theta\)?
Problem 8: A Pivotal Quantity
Next, determine a pivotal quantity for \(\theta\) and use that to calculate the endpoints of a \((1-\alpha) \times 100\%\) confidence interval for \(\theta\).
Problem 9: Jerzy and Egon: Neyman and Pearson Next Generations*
Finally, use the Neyman–Pearson Lemma to find the value of K for testing
\[
\begin{align*}
H_0:& \hspace{1ex} \theta = 1 \\
H_A:& \hspace{1ex} \theta = 4
\end{align*}
\]
For consistency, let us keep \(\alpha = 0.05\) and \(n = 4\) here. Let us also set \(m = 10\).
Yes, there is (almost) never a need to calculate the actual value of K. However, past experience tells me that connecting the Neyman-Pearson results to the foundations helps us better understand how it is really used. You will do a lot of algebra here, followed by a CDF calculation, followed by more algebra. Bonne chance !
The First Extra Credit Problem: The Uniform Connection
Let \(X \sim \text{UNIF}(0,1)\). Define \(Y = -\ln X \). Prove
\[
Y \sim \text{Exp}(1)
\]
This relation was used to generate random Exponential values back in the good old days.
The Second Extra Credit Problem: The Beta Connection
Prove that the Exponential distribution is a limit of a scaled Beta distribution. That is, prove
\[
\lim_{n \to \infty}\ n\ \text{BETA}(1,\ n) \stackrel{\text{dist}}{=} \text{Exp}(1)
\]
Needing to show that the probability density functions are equal should show you that mathematics is not dead. There is still a need for understanding the limit process from first-term calculus.
This page was last modified on 29 April 2025.