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Gaussian Random Vectors and Processes 3.1 Introduction 3.2 Gaussian Random Variables
| Content Provider | Semantic Scholar |
|---|---|
| Abstract | Poisson processes and Gaussian processes are similar in terms of their simplicity and beauty. When we first look at a new problem involving stochastic processes, we often start with insights from Poisson and/or Gaussian processes. Problems where queueing is a major factor tend to rely heavily on an understanding of Poisson processes, and those where noise is a major factor tend to rely heavily on Gaussian processes. Poisson and Gaussian processes share the characteristic that the results arising from them are so simple, well known, and powerful that people often forget how much the results depend on assumptions that are rarely satisfied perfectly in practice. At the same time, these assumptions are often approximately satisfied, so the results, if used with insight and care, are often useful. This chapter is aimed primarily at Gaussian processes, but starts with a study of Gaussian (normal 1) random variables and vectors, These initial topics are both important in their own right and also essential to an understanding of Gaussian processes. The material here is essentially independent of that on Poisson processes in Chapter 2. A random variable (rv) W is defined to be a normalized Gaussian rv if it has the density f W (w) = 1 p 2⇡ exp ✓ w 2 2 ◆ ; for all w 2 R. 1 Gaussian rv's are often called normal rv's. I prefer Gaussian, first because the corresponding processes are usually called Gaussian, second because Gaussian rv's (which have arbitrary means and variances) are often normalized to zero mean and unit variance, and third, because calling them normal gives the false impression that other rv's are abnormal. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://www.rle.mit.edu/rgallager/documents/6.262lateweb3.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
