The Premium Pen Factory tests one pen in each batch of 100...

The Premium Pen Factory tests one pen in each batch of 100 pens. The ink-filling machine is bipolar, so pens can write continuously for an exponential duration of mean either $1 / 2$ hour or 5 hours. The machine is in the short-life production mode $10 \%$ of the time. A batch of short-life pens sold as long-life pens results in a loss of $$\$ 5$$, while a batch of long-life pens mistakenly sold as short-life results in a loss of $$\$ 3$$. Find the Bayes decision rule to decide whether a batch is long-life or short-life based on the measured lifetime of the test pen.

Key Concepts

Bayesian Decision Theory

This concept involves making decisions under uncertainty by using probabilities to combine prior information and the likelihood of observed data. In decision theory, actions are chosen based on minimizing expected losses (or maximizing utility) corresponding to possible true states of the world, integrating cost implications with uncertainty about outcomes.

Exponential Distribution

The exponential distribution is a continuous probability distribution often used to model the time until an event occurs, such as the lifetime of a product. It is characterized by a constant hazard rate, making it memoryless and useful in reliability analysis and survival studies.

Loss Function

A loss function quantifies the penalty or cost associated with making incorrect decisions in a statistical inference problem. In contexts like hypothesis testing, different kinds of misclassification errors are assigned different penalties, and the optimal decision rule minimizes the expected loss (or risk) based on both the probabilities and the costs.

Likelihood Ratio Test

The likelihood ratio test is a statistical method used to compare the evidences provided by data under two competing hypotheses. In Bayesian decision-making, this test helps in determining thresholds for decisions by comparing the ratio of the probabilities (densities) of observed data under each hypothesis, effectively incorporating both the data's information and prior probabilities into the decision process.

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