Fundamentals of Probability: A First Course pp 127-169 | Cite as

# Continuous Random Variables

- 6.5k Downloads

## Abstract

We mentioned in Chapter 4 that discrete random variables serve as good examples to develop probabilistic intuition, but they do not account for all the random variables that one studies in theory and applications. In this chapter, we introduce the so-called *continuous random variables*, which typically take all values in some nonempty interval; e.g., the unit interval, the entire real line, etc. The right probabilistic paradigm for continuous variables cannot be pmfs. Discrete probability, which is based on summing things, is replaced by integration when we deal with continuous random variables and, instead of pmfs, we operate with a density function for the variable The density function fully describes the distribution, and calculus occupies the place of discrete operations, such as sums, when we come to continuous random variables. The basic concepts and examples that illustrate how to do the basic calculations are discussed in this chapter. Distinguished continuous distributions that arise frequently in applications will be treated separately in later chapters.

## Keywords

Density Function Hazard Rate Moment Generate Function Discrete Random Variable Continuous Random Variable## Preview

Unable to display preview. Download preview PDF.

## References

- Bernstein, S. (1947).
*Theory of Probability*(Russian), Moscow Leninghad.Google Scholar - Bucklew, J. (2004).
*Introduction to Rare Event Simulation*, Springer, New York.zbMATHGoogle Scholar - Chernoff, H. (1952). A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations, Ann. Math. Statist., 23, 493–507.zbMATHCrossRefMathSciNetGoogle Scholar
- DasGupta, A. (2008).
*Asymptotic Theory of Statistics and Probability*, Springer, New York.zbMATHGoogle Scholar - Dembo, A. and Zeitouni, O. (1998).
*Large Deviations, Techniques and Applications*, Springer, New York.zbMATHGoogle Scholar - den Hollander, F. (2000).
*Large Deviations*, Fields Institute Monograph, AMS, Providence, RI.zbMATHGoogle Scholar - Lugosi, G. (2006).
*Concentration of Measure Inequalities*, Lecture Notes, Dept. of Economics, Pompeu Fabra University., Barcelona.Google Scholar - Varadhan, S.R.S. (2003).
*Large Deviations and Entropy*, Princeton University Press, Princeton, NJ.Google Scholar