MATHEMATICAL PRELIMINARIES Independent Component Analysis. Aapo Hyv¨ rinen, Juha Karhunen, Erkki Oja a Copyright 2001 John Wiley & Sons, Inc. ISBNs: 0-471-40540-X (Hardback); 0-471-22131-7 (Electronic) 2 Random Vectors and Independence In this chapter, we review central concepts of probability theory,statistics, and random processes. The emphasis is on multivariate statistics and random vectors. Matters that will be needed later in this book are discussed in more detail, including, for example, statistical independence and higher-order statistics. The reader is assumed to have basic knowledge on single variable probability theory, so that fundamental definitions such as probability, elementary events, and random variables are familiar. Readers who. | Independent Component Analysis. Aapo Hyvarinen Juha Karhunen Erkki Oja Copyright 2001 John Wiley Sons Inc. ISBNs 0-471-40540-X Hardback 0-471-22131-7 Electronic Part I MATHEMATICAL PRELIMINARIES Independent Component Analysis. Aapo Hyvarinen Juha Karhunen Erkki Oja Copyright 2001 John Wiley Sons Inc. ISBNs 0-471-40540-X Hardback 0-471-22131-7 Electronic _2 Random Vectors and Independence In this chapter we review central concepts of probability theory statistics and random processes. The emphasis is on multivariate statistics and random vectors. Matters that will be needed later in this book are discussed in more detail including for example statistical independence and higher-order statistics. The reader is assumed to have basic knowledge on single variable probability theory so that fundamental definitions such as probability elementary events and random variables are familiar. Readers who already have a good knowledge of multivariate statistics can skip most of this chapter. For those who need a more extensive review or more information on advanced matters many good textbooks ranging from elementary ones to advanced treatments exist. A widely used textbook covering probability random variables and stochastic processes is 353 . PROBABILITY DISTRIBUTIONS AND DENSITIES Distribution of a random variable In this book we assume that random variables are continuous-valued unless stated otherwise. The cumulative distribution function cdf Fx of a random variable .r at point x .t o is defined as the probability that x x0 Fx x0 P x x0 Allowing x0 to change from oo to oo defines the whole cdf for all values . Clearly for continuous random variables the cdf is a nonnegative nondecreasing often monotonically increasing continuous function whose values lie in the interval 15 16 RANDOM VECTORS AND INDEPENDENCE A gaussian probability density function with mean m and standard deviation r. 0 Fx x 1. From the definition it also follows directly that
