ABSTRACT
The previous chapter essentially dealt with one-dimensional random variables and their probabilistic characterization and properties. Frequently a joint probabilistic analysis of two or more random variables is necessary. For instance, for weather predictions the meteorologist must take into account the interplay of randomly fluctuating parameters as air pressure, temperature, wind force and direction, humidity, et cetera. The operator of a coal power station, in order to be able to properly planning the output of the station, needs to take into account outdoor temperature as well as ash and water content of the coal presently available. These three parameters have a random component and there is a dependency between ash and water content. The information technologist, when analyzing stochastic signals, has jointly to consider their random phases and amplitudes. The forester, who has to estimate the amount of wood in a forest stand, measures both height and stem diameter (at a height of 1.3 m) of trees. Even in chapter 2 of this book vectors of random variables occurred without having explicitely hinted to this: When a die is tossed twice, then the outcome is (X 1, X 2). The binomial distribution is derived from a sequence of n binary random variables (X 1, X 2, …, Xn ).More challenging situations will be discussed in Part II of this book: Let, for instance, X(t)be the price of a unit of stock at time t and 0 < t 1 < t 2 < … < tn . Then the components of the n-dimensional vector (X(t 1), X(t 2), …, X(tn ))are the random stock prices at time points ti . There is an obvious dependency between the X(ti )so that for the prediction of the stock price development in time the random variables X(ti )should not be analyzed separately of each other. The same refers to other time series as registering temperatures, population sizes, et cetera, at increasing time points.
