 # Probabilistic Analysis

DOI: 10.4018/978-1-4666-8315-0.ch001
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## Abstract

This chapter provides the motivations behind the usage of probabilistic analysis in the domains of science and engineering. This is followed by a brief introduction of some of the foremost concepts of probabilistic analysis and the widely used probabilistic analysis techniques. The chapter concludes by highlighting some of the limitations of the traditional probabilistic analysis techniques.
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## 1.1 Motivation

It is remarkable that this science, which originated in the consideration of games of chance, should have become the most important object of human knowledge … The most important questions of life are, for the most part, really only problems of probability. (Pierre-Simon, Marquis de Laplace [1749-1827])

This quote by the famous French mathematician and astronomer may appear exaggerated, but it is a fact that probabilistic analysis has become a tool of fundamental importance in almost every area of science and engineering. A system may be purely deterministic but the unpredictable and random nature of its real-world surroundings makes it very hard to predict its exact outputs. The randomness may originate from the unwanted noise effects, failures due to aging of the components of an engineering system or the unpredictable traffic pattern in the case of telecommunication protocols. Due to these random components, establishing the correctness of a system under all circumstances usually becomes impractically expensive. For example, consider a simple telecommunication protocol, with handshake, that is used to ensure reliable communication in the presence of noise. We can never formally verify that the packet sent for transmission would be received properly because there is a possibility that the noisy channel may corrupt the transmission all the time. Probabilistic analysis is thus used. The main idea behind probabilistic analysis is to mathematically model the random and unpredictable elements of the given system and its environment by appropriate random variables. The probabilistic properties of these random variables are then used to judge the system's behavior regarding parameters of interest, such as downtime, availability, number of failures, capacity, and cost. Thus, instead of guaranteeing that the system meets some given specification under all circumstances, the probability that the system meets this specification is reported (Mitzenmacher & Upfal, 2005). Again considering the example of the handshake protocol, we can model the behavior of the noise with an appropriate random variable and thus use its corresponding probabilistic and statistical properties to judge the probability of successful transmission.

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## 1.2 Randomized Models

The randomized models of systems, which exhibit uncertainties, are constructed using appropriate random variables. For example, a random variable may be used to mathematically describe the outcome of a coin or a die. A random variable can be defined mathematically as a function that maps elements of a probability space to some other measurable space. Random variables can be broadly classified into two types, i.e., discrete and continuous.

A random variable is called discrete if the number of different values that it can acquire, or its range, is finite or at most countably infinite. For example, a coin toss and roll of a die can be modeled using discrete random variables. Discrete random variables can be used to construct randomized models of many real-world systems. For example, channel noise in digital communication is usually modeled by the Bernoulli random variable, randomized algorithms often use the Geometric random variable and the Binomial random variable has numerous applications in quality control problems.

A random variable is called continuous if it can attain an infinite number of values or in other words its range is an interval of real numbers. For example, the arrival time of a data packet at a web server and the temperature of an electrical component can be modeled using continuous random variables. Many real-world systems exhibit randomness of continuous nature and thus continuous random variables are used to construct their randomized models in probabilistic analysis. For example, the Continuous Uniform distribution is used to model quantization errors in computer arithmetic applications, the Exponential distribution occurs in queuing applications to model inter arrival and service times and the Normal distribution is extensively used to model signals in data transmission and digital signal processing systems.

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