ABSTRACT
We frequently have the occasion to study the mutual interactions between two or more variables. How useful are a student’s college grade point average (GPA) and score on the law school admissions test (LSAT) as predictor variables for a student’s future performance in law school? How does diversification of an investment portfolio reduce risk, and how does it affect the returns? For example, when oil prices soar so do the stock prices of oil companies, while those of automobile manufacturers and airlines fall. Modern portfolio theory begins with an analysis of the joint variation of the different asset classes (such as stocks, bonds, and mutual funds) in the portfolio. In environmental studies the air quality of a metropolitan region is determined by simultaneously measuring several pollutants such as ozone, nitrogen oxide, hydrocarbons, etc. Before undertaking a plan to reduce the pollution levels the environmental scientist must study their interactions since reducing the level of just one pollutant could actually increase the combined harmful effects of the others. Statisticians use the joint distribution function to study the joint variations among the variables. We begin with the discrete case and follow with the continuous case, the only difference between them is that, in the latter case, we use some basic tools from multivariate calculus.
