Problem 52 - Applied Statistics and Probability for Engineers

Question

Let \(X\) and \(Y\) represent the concentration and viscosity of a chemical product. Suppose that \(X\) and \(Y\) have a bivariate normal distribution with \(\sigma_{X}=4, \sigma_{Y}=1, \mu_{X}=2\) and \(\mu_{y}=1\). Draw a rough contour plot of the joint probability density function for each of the following values of \(\rho\) : (a) \(\rho=0\) (b) \(\rho=0.8\) (c) \(\rho=-0.8\)

Step-by-Step Solution

Answer

Contour plots are concentric ellipses centered at \((2, 1)\); they align parallel to coordinate axes for \(\rho=0\), tilt to upper-right for \(\rho=0.8\), and tilt to upper-left for \(\rho=-0.8\).

Step 1: Understand the Bivariate Normal Distribution

A bivariate normal distribution describes the joint distribution of two variables, say \(X\) and \(Y\), where each variable is normally distributed. The distribution is characterized by the means \(\mu_X\) and \(\mu_Y\), standard deviations \(\sigma_X\) and \(\sigma_Y\), and the correlation coefficient \(\rho\).

Step 2: Identify Key Parameters

The given parameters are:- Means: \(\mu_X = 2\), \(\mu_Y = 1\).- Standard deviations: \(\sigma_X = 4\), \(\sigma_Y = 1\).- Correlation \( \rho \) will be different for parts (a), (b), and (c).

Step 3: Setting Up the Contour Plot

In a contour plot, ellipses represent levels of equal joint probability density. The centers of these ellipses are at \((\mu_X, \mu_Y) = (2, 1)\). The shapes and orientations of the ellipses depend on the value of \(\rho\).

Step 4: Case (a) – When \(\rho = 0\)

When \(\rho = 0\), \(X\) and \(Y\) are uncorrelated. The contour ellipses are regular, with axes aligned with the coordinate axes. The ellipse is centered at \( (2, 1) \) with primary axes paralleling \(X\) and \(Y\).

Step 5: Case (b) – When \(\rho = 0.8\)

When \(\rho = 0.8\), \(X\) and \(Y\) are positively correlated. The contour ellipses tilt towards the upper-right, reflecting this positive relationship. These ellipses elongate along the line \(Y = X\) due to high correlation.

Step 6: Case (c) – When \(\rho = -0.8\)

When \(\rho = -0.8\), \(X\) and \(Y\) are negatively correlated. The contour ellipses tilt towards the upper-left, showing a negative relationship. Ellipses elongate along the line \(Y = -X\) to reflect this correlation.

Key Concepts

Correlation Coefficient

In the context of a bivariate normal distribution, the correlation coefficient, often denoted by \( \rho \), measures the strength and direction of the linear relationship between two variables \(X\) and \(Y\). It is a vital parameter when understanding the interaction between variables. Correlation coefficients can range from -1 to 1.

• A correlation of \( \rho = 1 \) signifies a perfect positive linear relationship, meaning as one variable increases, the other one does as well.
• Conversely, \( \rho = -1 \) indicates a perfect negative linear relationship; as one variable increases, the other decreases.
• When \( \rho = 0 \), it suggests there is no linear relationship between \(X\) and \(Y\), though they might still have a non-linear association.

In a real-world bivariate normal distribution, these movements help in predicting how changes in one variable might affect the other.

Contour Plot

A contour plot is a graphical representation where lines connect points of equal value. For a bivariate normal distribution, these lines commonly form ellipses. These ellipses indicate regions of equal joint probability density, effectively visualizing the relationship and behavior of two variables together.

• In the case of \( \rho = 0 \), the contour lines are circular ellipses with their axes aligned to the coordinate axes, reflecting no correlation.
• With \( \rho = 0.8 \), the ellipses become elongated and tilt towards an upward right direction, portraying a strong positive linear association.
• For \( \rho = -0.8 \), you'll notice the ellipses tilting toward the upper-left, illustrating a significant negative relationship.

Such visualizing tools are valuable for understanding and predicting the joint behavior of the variables, providing insight at a glance.

Joint Probability Density Function

The joint probability density function (PDF) describes the likelihood of two continuous random variables \(X\) and \(Y\) simultaneously taking on a specific pair of values. This function is crucial to understanding how two quantities will behave together under a bivariate normal distribution.

• For a given pair \((x, y)\), the joint PDF of \(X\) and \(Y\) is determined by their individual means and standard deviations, as well as the correlation coefficient \( \rho \).
• The joint PDF forms the basis for the contour plot, dictating how flat or peaked the probability surface is over different regions.

The joint PDF helps in explaining phenomena where two variables don't just act separately but have interdependent behaviors, offering a statistical lens into their dynamics.