Question

Let \(X\) and \(Y\) have the joint pdf $$ f\left(x, y ; \theta_{1}, \theta_{2}\right)=\frac{1}{\theta_{1} \theta_{2}} \exp \left(-\frac{x}{\theta_{1}}-\frac{y}{\theta_{2}}\right), \quad 0

Short answer

The form of the classification rule is: classify \((x, y)\) as \((\theta_{1}'=1, \theta_{2}'=5)\) if \(5x + 2y < 16\) and classify \((x, y)\) as \((\theta_{1}''=3, \theta_{2}''=2)\) if \(5x + 2y > 16\).

Step-by-step solution

Formulate the two pdfs based on the given conditions

Substitute \((\theta_{1}'=1, \theta_{2}'=5)\) and \((\theta_{1}''=3, \theta_{2}''=2)\) into the joint pdf to obtain the pdfs for each condition. \n For \((\theta_{1}'=1, \theta_{2}'=5)\) the pdf will be: \[ f(x,y; 1, 5) = e^{-x - \frac{y}{5}} \] and for \((\theta_{1}''=3, \theta_{2}''=2)\) the pdf will be: \[ f(x,y; 3, 2) = \frac{1}{6}e^{-\frac{x}{3} - \frac{y}{2}} \]

Formulate the classification rule

The classification rule generally classifies a given observation to the condition where its probability based on the pdf is higher. Hence, the classification rule will be of the form: Classify the observed \((x, y)\) to the condition \((\theta_{1}'=1, \theta_{2}'=5)\) if \[ f(x, y; 1, 5) > f(x, y; 3, 2) \] and classify the observed \((x, y)\) to the condition \((\theta_{1}''=3, \theta_{2}''=2)\) if \[ f(x, y; 1, 5) < f(x, y; 3, 2) \]

Simplify the classification rule

Simplify the inequalities obtained in step 2 to get a simplified condition using which one can easily classify the observation. On simplifying you will arrive at: Classify \((x, y)\) as \((\theta_{1}'=1, \theta_{2}'=5)\) if \[ 5x+2y < 16 \] and classify \((x, y)\) as \((\theta_{1}''=3, \theta_{2}''=2)\) if \[ 5x + 2y > 16 \]

Key concepts

Joint Probability Density Function

The joint probability density function (pdf) is a mathematical description that defines the likelihood of two continuous random variables occurring simultaneously. In our exercise scenario, the random variables are denoted as X and Y, each depending on parameters \theta_1 and \theta_2. The joint pdf is represented by the following equation:
\[ f(x, y ; \theta_1, \theta_2)=\frac{1}{\theta_1 \theta_2} \exp \left(-\frac{x}{\theta_1}-\frac{y}{\theta_2}\right), \quad 0This function only holds value when x and y are greater than zero, reflecting the domain of the variables. Otherwise, the pdf is zero. Joint pdfs are essential in multi-variable statistics, as they generalize the concept of a pdf from one to several dimensions, allowing us to model more complex randomness that involves multiple variables.
In the context of classification, the parameters of the pdf (\theta_1, \theta_2) are indicative of different conditions or classes. To compute the probability of an event under each class, one substitutes the respective parameters into the joint pdf equation.

Probability Density Function

A probability density function (pdf) describes the relative likelihood for a continuous random variable to take on a given value. The pdf differs from a probability mass function, which is used for discrete random variables; pdfs are for continuous data where probabilities for exact values are inherently zero since there are infinitely many possible values. Instead, the pdf provides the probability that the variable falls within a particular range of values. It is visualized as a curve where the total area under the curve corresponds to a probability of one.
When simplifying the pdf for computational purposes, as in our exercise, one must handle the exponential terms in a way that facilitates the classification process. Understanding the pdf shape and behavior according to its parameters is key in defining what are known as decision boundaries in classification tasks.

Statistical Decision Theory

Statistical decision theory involves techniques for making optimal decisions under uncertainty and is foundational in classification tasks. It incorporates principles of statistics and probability theory to assess the risks and benefits of different decisions, aiming to minimize error or loss. In the given exercise, we use statistical decision theory to formulate a rule for classifying an observation into one of two possible parameter conditions.
The classification rule is based on comparing probabilities calculated from the separate pdfs, each representing a unique class. It allocates an observation to the class with the higher probability (based on the estimated pdf). This process of comparison and decision-making is a practical application of statistical decision theory. A key step in this theory is simplifying complex decision rules into manageable formulas, such as linear inequalities that indicate when to classify an observation into a particular category. Here, after simplification, we get a line defined by \(5x + 2y = 16\), serving as the decision boundary.