Abstract
In any situation in which the outcome of an event is uncertain, we can talk about the probabilities of such outcomes, i.e. how likely they are. The most popular example is tossing of a fair coin: since a coin is fair and there are no other outcomes except the heads and tails, they are both equally probable and we say that their probabilities are eqaual to 1/2. As a branch of mathematics, the discipline of probability provides formal methods for quantifying chances of such uncertain outcomes. In this chapter, we first introduce the basic probability concepts, and show how to calculate the probabilities of some events using the rules from probability theory. Next, we present the concept of a random variable which enable us to move from the event outcomes themselves to a numerical function of that outcomes. For example, we can define a random variable X, being the number of heads we get from tossing two coins: X could then be 0, 1, or 2 heads. Except for such discrete random variable, the continuous one can take any value in some interval on the real number line. We characterize here the probability distributions (i.e. the probability with which they take on the various values in their range) of most representative random variables. Then, we also discuss probability models for the joint (simultaneous) behaviour of two or more random variables which is important for describing their dependency. At the end, after presenting the central limit theorem which states about limit distribution of sample mean, the most important theorem for statistical analysis, we show how to apply computer simulation methods to probability problems and model random events.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Bibliography
Billingsley, P.: Probability and Measure, 3rd edn. Wiley, New York (1995)
Hodges Jr., J.L., Lehmann, E.L.: Basic Concepts of Probability and Statistics, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia (2005)
Papoulis, A., Pillai, S.U.: Probability, Random Variables and Stochastic Processes, 4th edn. McGraw-Hill, New York (2002)
R Core Team: R language definition. http://cran.r-project.org/doc/manuals/r-release/R-lang.pdf (2019). Accessed 16 Dec 2019
Rudin, W.: Principles of Mathematical Analysis, 3rd edn. McGraw-Hill, New York (1976)
Soong, T.T.: Fundamentals of Probability and Statistics for Engineers, 1st edn. Wiley, Chichester (2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Stapor, K. (2020). Elements of Probability Theory. In: Introduction to Probabilistic and Statistical Methods with Examples in R . Intelligent Systems Reference Library, vol 176. Springer, Cham. https://doi.org/10.1007/978-3-030-45799-0_1
Download citation
DOI: https://doi.org/10.1007/978-3-030-45799-0_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-45798-3
Online ISBN: 978-3-030-45799-0
eBook Packages: EngineeringEngineering (R0)