Random Signals: Statistics Basis

Abstract

This chapter recalls the basis of the statistics for a random variable with values in a continuum. We define the probability density function and the cumulative distribution function of a r.v.: expectancy, variance, the moments of a distribution, and the characteristic function. Particular attention is paid to the Gaussian distribution (normal distribution). The probability density function of a function of a random variable is determined. In the second part of this chapter, we present the statistics of two random variables. We define their joint probability density and the marginal probability densities. We give an overview of the Bayesian statistical aspect. We define the correlation coefficient, the orthogonality and the independence in probability. These concepts are used for the study of two joined Gaussian variables. It is then shown that the probability density function of the sum of two independent r.v. is the convolution of their probability densities. This result is extended qualitatively to the sum of a large number of independent random variables that appear to follow approximately a Gaussian distribution. This result is known as the central limit theorem. We finally consider the statistical distribution of complex variables and the correlation of two complex r.v. A table of Gauss’s law is given at the end of the chapter.