# Integer-Valued and Discrete Random Variables

Chapter
Part of the Springer Texts in Statistics book series (STS)

## Abstract

In this chapter, we introduce the concept of random variables and their distributions. In some sense, the entire subject of probability and statistics is about distributions of random variables. Random variables, as the very name suggests, are quantities that vary over time or from individual to individual, and the reason for the variability is some underlying random process. We try to understand the behavior of a random variable by analyzing the probability structure of that underlying random mechanismor process. Random variables, like probabilities, originated in gambling. Therefore, the random variables that come to us more naturally are integer-valued random variables; e.g., the sum of the two rolls when a die is rolled twice. Integervalued random variables are special cases of what are known as discrete random variables. We study integer-valued and discrete random variables and their basic properties in this chapter. Random variables with an intrinsically more abstract and complex structure will be studied after we introduce probability density functions.

## Keywords

Cumulative Distribution Function Independent Random Variable Sample Space Discrete Random Variable Continuous Random Variable
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## References

1. Alon, N. and Spencer, J. (2000). The Probabilistic Method, Wiley, New York.
2. Chow, Y. and Studden, W. (1969). Monotonicity of the variance under truncation and variations of Jensen’s inequality, Ann. Math. Statist., 40, 1106–1108.
3. Erdös, P. and Rényi, A. (1970). On a new law of large numbers, J. Anal. Math., 23, 103–111.
4. Erdös, P. and Révész, P. (1975). On the length of the longest head-run, Colloq. Janos Bolyai Math. Soc., Topics Inform. Theory I., 16, 219–228.Google Scholar
5. Paley, R.E. and Zygmund, A. (1932). A note on analytic functions in the unit circle, Proc. Cambridge Philos. Soc., 28, 266–272.
6. Rao, C.R. (1973). Linear Statistical Inference and Applications, Wiley, New York.