In previous chapters, we examined samples and populations with respect to one variable or attribute only. That is, we restricted ourselves to one-dimensional analysis.^{[40]} However, for many applications of statistics to problems in finance, there is typically less of a need to analyze one variable in isolation. Instead, a typical problem faced by practitioners is to investigate the common behavior of several variables and joint occurrences of events. In other words, there is the need to establish joint frequency distributions. Along with measures determining the extent of dependence between variables, we will also introduce graphical tools for higher dimensional data to obtain a visual conception of the underlying data structure.

As in the one-dimensional case, we first gather all joint observations of our variables of interest. For a better overview of occurrences of the variables, it might be helpful to set up a table with rows indicating observations and columns representing the different variables. This table is called the table of observations. Thus, the cell of, say, row *i* and column *j* contains the value that observation *i* has with respect to variable *j*. Let us express this relationship between observations and variables a little more formally by some functional representation.

In the following, we will restrict ourselves to observations of pairs, that is, *k* = 2. In this case, the observations are ...

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