Density Functions of Multiple Random Variables

So far, we have mainly discussed probability (density) functions of single random variables. As mentioned earlier, we use random variables to describe data such as sensor readings in robots. It’s important to note that one robot typically has several sensors. Thus, in many applications, we consider multiple random variables. The quantities described by the random variables might be independent but, in many cases, they are also related. Indeed, we will later talk about how to describe various types of relations. Thus, in order to discuss situations with multiple random variables, or multivariate statistics, it is useful to know the basic rules.

Types of distribution

Stochastic machine learning models usually contain many random variables. Here, we introduce the multivariate concepts for two variables, although these concepts readily generalize directly to an arbitrary number of variables. We start with some essential definitions. The total knowledge about the co-occurrence of specific values for two random variables X and Y is captured by the following equation: