Introduction to Probability Theory

By S. Bandyopadhyay, R. Bhattacharya. 2014

100 citations recorded by the source corpus.

Any realistic model of a real-world phenomenon must take into account the possibility of randomness. That is, more often than not, the quantities we are interested in will not be predictable in advance but, rather, will exhibit an inherent variation that should be taken into account by the model. This is usually accomplished by allowing the model to be probabilistic in nature. Such a model is, naturally enough, referred to as a probability model. The majority of the chapters of this book will be concerned with different probability models of natural phenomena. Clearly, in order to master both the " model building " and the subsequent analysis of these models, we must have a certain knowledge of basic probability theory. The remainder of this chapter, as well as the next two chapters, will be concerned with a study of this subject. Suppose that we are about to perform an experiment whose outcome is not predictable in advance. However, while the outcome of the experiment will not be known in advance, let us suppose that the set of all possible outcomes is known. This set of all possible outcomes of an experiment is known as the sample space of the experiment and is denoted by S. Some examples are the following. 1. If the experiment consists of the flipping of a coin, then S = {H, T } where H means that the outcome of the toss is a head and T that it is a tail.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/c2ec33874743672c992d51d9dcdd26299454f87c

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

In Statistics and Probability

Citations in the corpus

Communication dynamics in complex brain networks (2017, background): “Formally, these properties are captured by a pairwise statistic called the mean first-passage time, which measures the expected length of a (random) walk (in terms of number of traversed edges) between two nodes 113 .”
Variational Implicit Processes (2018, background): “Here (m)n = m(xn) and (Kff )n,n′ = K(xn,xn′), and following Kolmogorov consistency theorem (Itô, 1984), the mean and covariance functions m(·), K(·, ·) are shared across all such finite collections.”