A unique approach to stochastic processes that connects themathematical formulation of random processes to their use inapplications
This book presents an innovative approach to teachingprobability theory and stochastic processes based on the binaryexpansion of the unit interval. Departing from standard pedagogy, it uses the binary expansion of the unit interval to explicitlyconstruct an infinite sequence of independent random variables (ofany given distribution) on a single probability space. Thisconstruction then provides the framework to understand themathematical formulation of probability theory for its use inapplications.
- The theory is presented first for countable sample spaces(Chapters 1-3) and then for uncountable sample spaces (Chapters4-18)
- Coverage of the explicit construction of i.i.d. randomvariables on a single probability space to explain why it is thedistribution function rather than the functional form of randomvariables that matters when it comes to modeling randomphenomena
- Explicit construction of continuous random variables tofacilitate the "digestion" of random variables, i.e., how they areused in contrast to how they are defined
- Explicit construction of continuous random variables tofacilitate the two views of expectation: as integration overthe underlying probability space (abstract view) or as integrationusing the density function (usual view)
- A discussion of the connections between Bernoulli, geometric, and Poisson processes
- Incorporation of the Johnson-Nyquist noise model and anexplanation of why (and when) it is valid to use a delta functionto model its autocovariance
Comprehensive, astute, and practical, Introduction toProbability Theory and Stochastic Processes is a clearpresentation of essential topics for those studying communications, control, machine learning, digital signal processing, computernetworks, pattern recognition, image processing, and codingtheory.