Uniform distribution, the simplest probability distribution, plays an important role in Statistics since it is indispensable in modeling random variables, and therefore in traditional and Quasi-Monte Carlo simulation. It is often used to represent the distribution of roundoff errors in values tabulated to the nearest k decimal places (Johnson et al. 1995). We can distinguish between the continuous and discrete uniform distribution.
Properties of the Uniform Distribution
The continuous random variable X is said to be uniformly distributed, or having rectangular distribution on the interval [a, b], and we write X : U(a, b), if its probability density function (p.d.f) equals \(f(x) = \frac{1} {b-a},\ x \in \left [a,b\right ],\) and 0 elsewhere. It follows that the distribution function is \(F(x) = \frac{x-a} {b-a} ,\ x \in \left [a,b\right ].\) The moments are \({m}_{r} = \frac{1} {r+1} \frac{{b}^{r+1}-{a}^{r+1}} {b-a} ,\ r \in N,\) while the central moments are \({\mu }_{2k-1} = 0,\ {\mu...
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References and Further Reading
Sobol I (1973) islenn e metod Monte Karlo, Nauka, Moskva (in Russian)
Dexter CW, Hogg RV (2001) A little uniform density with big instructional potential. J Stat Educ 9(2)
Djorić D, Jevremović V et al (2007) Atlas raspodela. Gradjevinski fakultet, Beograd (in Serbian)
Hogg RV, McKean JW, Craig AT (2005) Introduction to mathematical statistics. Pearson Education International, Upper Saddle River
Johnson NL, Kotz S, Balakrishnan N (1995) Continuous univariate distributions, vol 2, 2nd edn. Wiley-Interscience, New York
Romano JP, Siegel AF (1986) Counterexamples in probability and statistics. Chapman & Hall/CRC Press, New York
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Jevremović, V. (2011). Uniform Distribution in Statistics. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_642
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