Abstract
Whether an experiment yields qualitative or quantitative outcomes, methods of statistical analysis require that we focus on certain numerical aspects of the data (such as a sample proportion x/n, mean \( \bar{x} \), or standard deviation s). The concept of a random variable allows us to pass from the experimental outcomes themselves to a numerical function of the outcomes. There are two fundamentally different types of random variables—discrete random variables and continuous random variables. In this chapter, we examine the basic properties and discuss the most important examples of discrete variables. Chapter 4 focuses on continuous random variables.
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- 1.
P(X = x) is read “the probability that the rv X assumes the value x.” For example, P(X = 2) denotes the probability that the resulting X value is 2.
- 2.
“Between a and b, inclusive” is equivalent to (a ≤ X ≤ b).
- 3.
A quantity is o(Δt) (read “little o of delta t”) if, as Δt approaches 0, so does o(Δt)/ Δt. That is, o(Δt) is even more negligible than Δt itself. The quantity (Δt)2 has this property, but sin(Δt) does not.
Bibliography
Durrett, Richard, Elementary Probability for Applications, Cambridge Univ. Press, London, England, 2009.
Johnson, Norman, Samuel Kotz, and Adrienne Kemp, Univariate Discrete Distributions (3rd ed.), Wiley-Interscience, New York, 2005. An encyclopedia of information on discrete distributions.
Olkin, Ingram, Cyrus Derman, and Leon Gleser, Probability Models and Applications (2nd ed.), Macmillan, New York, 1994. Contains an in-depth discussion of both general properties of discrete and continuous distributions and results for specific distributions.
Pitman, Jim, Probability, Springer-Verlag, New York, 1993.
Ross, Sheldon, Introduction to Probability Models (9th ed.), Academic Press, New York, 2006. A good source of material on the Poisson process and generalizations and a nice introduction to other topics in applied probability.
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Devore, J.L., Berk, K.N. (2012). Discrete Random Variables and Probability Distributions. In: Modern Mathematical Statistics with Applications. Springer Texts in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0391-3_3
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DOI: https://doi.org/10.1007/978-1-4614-0391-3_3
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