Question

Let \(X\) and \(Y\) have a joint bivariate normal distribution. An observation \((x, y)\) arises from the joint distribution with parameters equal to either $$ \mu_{1}^{\prime}=\mu_{2}^{\prime}=0, \quad\left(\sigma_{1}^{2}\right)^{\prime}=\left(\sigma_{2}^{2}\right)^{\prime}=1, \quad \rho^{\prime}=\frac{1}{2} $$ Ior $$ \mu_{1}^{\prime \prime}=\mu_{2}^{\prime \prime}=1, \quad\left(\sigma_{1}^{2}\right)^{\prime \prime}=4, \quad\left(\sigma_{2}^{2}\right)^{\prime \prime}=9, \quad \rho^{\prime \prime}=\frac{1}{2} \text { . } $$ Show that the classification rule involves a second-degree polynomial in \(x\) and \(y\).

Short answer

The classification rule for the given bivariate distribution problem involves deriving the densities for the given parameters, and then equating these two densities. This results in a second degree polynomial, thus proving that the classification rule involves a second-degree polynomial in \(x\) and \(y\).

Step-by-step solution

Define the Densities

The probability density function of a bivariate distribution is given by the formula: \[f_{XY}(x,y) = \frac{1}{2π\sigma_{1}\sigma_{2}\sqrt{1-\rho^2}} e^{-\frac{1}{2(1-\rho^2)}} [(x-\mu_{1})^2/\sigma_{1}^2 - 2ρ(x-\mu_{1})(y-\mu_{2})/(\sigma_{1}\sigma_{2}) + (y-\mu_{2})^2/\sigma_{2}^2]\] Using this formula, we derive the densities for the two cases given in the problem. Let\[f_{XY}''(x,y)\] be the density when parameters are \(\mu_{1}'', \mu_{2}'', (\sigma_{1}^2)'', (\sigma_{2}^2)'', \rho''\) and \[f_{XY}'(x,y)\] be the density when parameters are \(\mu_{1}', \mu_{2}', (\sigma_{1}^2)', (\sigma_{2}^2)', \rho'\).

Equate the Two Densities

In order to find the classification rule, we equate the two densities derived above, \(f_{XY}''(x,y) = f_{XY}'(x,y)\)

Simplify the Equation in Step 2

We cancel common terms in the resulting equation, and divide both sides by the same common scalars, giving us a second-degree polynomial in \(x\) and \(y\) which is our required classification rule.

Key concepts

Probability Density Function

The probability density function (PDF) is a cornerstone concept in the realm of statistics and probabilistics, forming the bedrock of understanding various distributions, including the bivariate normal distribution. In essence, the PDF is a function that describes the likelihood of a random variable taking on a particular value. Imagine it as a way to map out the possibilities of different outcomes.

For a bivariate normal distribution, which encompasses two variables, say, X and Y, the PDF becomes a bit more complex. It accounts for both the individual variances of X and Y, denoted by \( \sigma_{1}^2 \) and \( \sigma_{2}^2 \) respectively, and their correlation \( \rho \).

The formula for the PDF of this intricate dance is given by: \[ f_{XY}(x,y) = \frac{1}{2\pi\sigma_{1}\sigma_{2}\sqrt{1-\rho^2}} e^{-\frac{1}{2(1-\rho^2)}} \left[\frac{(x-\mu_{1})^2}{\sigma_{1}^2} - \frac{2\rho(x-\mu_{1})(y-\mu_{2})}{(\sigma_{1}\sigma_{2})} + \frac{(y-\mu_{2})^2}{\sigma_{2}^2}\right] \]

In a specific scenario where the mean and variances of X and Y take on different values, we get separate PDFs. These functions express the probability distributions for the separate scenarios under consideration. Understanding the shape and nature of these functions is essential for students wishing to know how likely it is for given values of X and Y to occur under each scenario.

Classification Rule

The classification rule is a decision-making tool used to correctly identify to which population or group a new observation should belong based on existing data. Think of it as a guideline that helps to categorize new data points by comparing them against predetermined criteria.

In the context of the bivariate normal distribution exercise from the textbook, the classification rule assists in determining whether an observation with coordinates \(x, y\) arises from the first distribution with parameters \(\mu_{1}', \mu_{2}', \sigma_{1}^2', \sigma_{2}^2', \rho'\) or the second distribution with parameters \(\mu_{1}'', \mu_{2}'', \sigma_{1}^2'', \sigma_{2}^2'', \rho''\).

To find the classification rule, we equate the PDFs for the given parameters. Through this process, the equation you get after equating the two PDFs and simplifying will be a second-degree polynomial in x and y. The terms in this polynomial will include \( x^2 \) and \( y^2 \) but also could involve terms like \( xy \) given that \( \rho \) is not zero, and it accounts for the interaction between X and Y. Hence, in the simplification process, you'll get coefficients that are functions of the means, variances, and correlation parameters from the two scenarios.

Second-Degree Polynomial

A second-degree polynomial is a mathematical expression that features variables raised to no higher power than two. Represented generally as \( ax^2 + bxy + cy^2 + dx + ey + f \), it's a quadratic surface which could be a parabola, hyperbola, or ellipse depending on the coefficients a, b, and c.

In the step-by-step solution provided, after the PDFs are equated and simplified, the resulting function is indeed a second-degree polynomial in x and y. This polynomial is pivotal for the classification rule, allowing us to set boundaries between different scenarios.

Examining the coefficients of this polynomial sheds light on the nature and shape of the boundary. If the chosen values of x and y lead to a value from the polynomial equal to zero, this indicates the threshold between choosing one distribution over another for our classification rule. This threshold is a geometric curve on the xy-plane and affords a visual understanding of how data is segmented according to the rule.