Understanding the Expectation of a Bivariate Normal Distribution

In probability theory and statistics, the bivariate normal distribution is a fundamental concept, often used to model the joint distribution of two normally distributed random variables. This article aims to explore the intricacies of the bivariate normal distribution, particularly focusing on the calculation of the expectation of a given function of two such variables. This is not only crucial for advanced statistical analysis but also for fields such as econometrics, finance, and machine learning.

Introduction to Bivariate Normal Distribution

A bivariate normal distribution is characterized by two random variables (X) and (Y), which are jointly normally distributed. When (X) and (Y) follow a bivariate normal distribution, we can denote this using the notation ((X, Y) sim BN(mu_1, mu_2, sigma_1^2, sigma_2^2, rho)), where:

(mu_1) and (mu_2) are the means of (X) and (Y) respectively, (sigma_1^2) and (sigma_2^2) are the variances of (X) and (Y) respectively, (rho) is the correlation coefficient between (X) and (Y).

Calculating the Expectation

The expectation of the product of two jointly normally distributed random variables (X) and (Y) is given by the formula:

[E[XY] mu_1 rho cdot frac{sigma_1}{sigma_2} (Y - mu_2)]

This formula is a core concept in multivariate statistics and finds applications in various fields such as finance where correlation between two financial assets is crucial.

Solving a Specific Example

Let's consider a specific example to illustrate the application of the above formula. Suppose (X) and (Y) follow a bivariate normal distribution with the following parameters:

(mu_1 1), (mu_2 2), (sigma_1 4), (sigma_2 5), (rho frac{3}{4}).

Given the formula (E[XY] mu_1 rho cdot frac{sigma_1}{sigma_2} (Y - mu_2)), we substitute the specific values:

[E[XY] 1 cdot frac{3}{4} cdot frac{4}{5} (Y - 2)]

This simplifies to:

[E[XY] frac{3}{5} cdot 5 (Y - 2) 3 (Y - 2)]

So, the value of (E[XY]) is 7 when (Y 3).

Conclusion

The calculation of the expectation of a product of two normally distributed random variables is a fundamental task in multivariate statistics. Understanding and applying the bivariate normal distribution and its properties can significantly enhance the modeling capabilities in fields such as finance, economics, and machine learning.