Problem 10 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

Suppose we have a dataset \(x_{1}, \ldots, x_{n}\) that may be modeled as the realization of a random sample \(X_{1}, \ldots, X_{n}\) from an \(\operatorname{Exp}(\lambda)\) distribution, where \(\lambda\) is unknown. Let \(S_{n}=X_{1}+\cdots+X_{n}\). a. Check that \(\lambda S_{n}\) has a \(\operatorname{Gam}(n, 1)\) distribution. b. The following quantiles of the \(\operatorname{Gam}(20,1)\) distribution are given: \(q_{0.05}=\) \(13.25\) and \(q_{0.95}=27.88\). Use these to construct a \(90 \%\) confidence interval for \(\lambda\) when \(n=20\).

Step-by-Step Solution

Answer

\(\left(\frac{13.25}{s_n}, \frac{27.88}{s_n}\right)\) for \(\lambda\).

Step 1: Understanding the Relationship between Gamma and Exponential Distributions

If each random variable \(X_i\) is distributed as an \(\operatorname{Exp}(\lambda)\), the sum \(S_n = X_1 + X_2 + \cdots + X_n\) follows a \(\operatorname{Gamma}(n, \lambda)\) distribution. This is because a sum of \(n\) independent \(\operatorname{Exp}(\lambda)\) distributed variables is a \(\operatorname{Gam}(n, \lambda)\) distribution.

Step 2: Rescaling the Gamma Distribution

To show \(\lambda S_n\) has a \(\operatorname{Gam}(n, 1)\) distribution, note that rescaling a \(\operatorname{Gam}(n, \lambda)\) variable by its rate parameter \(\lambda\), transforms the rate to 1. Thus, \(\lambda S_n\) follows \(\operatorname{Gam}(n, 1)\).

Step 3: Using Quantiles to Find Confidence Interval for \(\lambda\)

Given quantiles for \(\operatorname{Gam}(20, 1)\), use \(q_{0.05}=13.25\) and \(q_{0.95}=27.88\) to find a 90% confidence interval for \(\lambda\). For the observed sum \(s_n\): \[ \Pr\left(13.25 < \lambda s_n < 27.88\right) = 0.9 \] implies: \[ \frac{13.25}{s_n} < \lambda < \frac{27.88}{s_n} \]. This is the desired 90% confidence interval.

Key Concepts

Exponential Distribution

An exponential distribution is a continuous probability distribution used to model the time between events in a Poisson process. This means it

• measures how much time until the next event occurs.
• is often used in scenarios like the time until a radioactive particle decays or the time until a call is received at a call center.

In mathematical terms, a variable is said to be exponentially distributed if it has the probability density function (PDF) \( f(x; \lambda) = \lambda e^{-\lambda x} \) for \( x \geq 0 \), where \( \lambda > 0 \) is the rate parameter which describes how frequently events occur.

The exponential distribution has a close relationship with the Poisson distribution. If you are randomly spacing events in time, the number of events that occur in an interval can be modeled as a Poisson distribution.

A key property of an exponentially distributed variable is memorylessness, meaning the probability of an event occurring in the future is independent of how much time has already passed.

Confidence Interval

A confidence interval is a range of values that is likely to contain a population parameter with a certain degree of confidence. It is a popular statistical tool used to estimate unknown parameters. Here, we focus on estimating the parameter \( \lambda \) of an exponential distribution.

To create this interval, researchers use sample data to calculate two numbers, forming a lower and upper bound within which the population parameter is expected to lie.

• The degree of confidence (like 90%) reflects how certain we are that the interval contains the parameter.
• Common confidence levels include 90%, 95%, and 99%.

The wider the interval means greater uncertainty about the parameter, while a narrow interval implies a more precise estimate.
In our exercise, we use the quantiles of the Gamma distribution to derive the confidence interval, taking the observed sum \( s_n \) into consideration, allowing us to determine intervals that are consistent with our level of confidence.

Random Sample

In statistics, a random sample is a subset of individuals chosen from a larger set (population). It serves as a fundamental concept in statistics that ensures each member of the population has an equal chance of being selected. This method avoids bias and allows for generalizations to be made about the population.

There are a few important aspects of a random sample:

• Each individual or data point is chosen randomly and entirely by chance.
• All possible samples have the same probability of being chosen.
• This procedure ensures the results can be generalized, representing the entire population.

Using random samples is critical when modeling data because the sample is assumed to represent the larger population. This is especially true when using distributions like the exponential distribution, where the properties derived from the sample data are supposed to estimate parameters like the rate \( \lambda \) accurately.
By ensuring the selection of a random sample, we enable more reliable and unbiased statistical inferences about populations, which can be particularly useful when constructing confidence intervals for unknown parameters.