The beta distribution is used for describing random variables having both lower and upper bounds. Random variables in hydrosystems that are bounded on both limits include reservoir storage and groundwater table for unconfined aquifers.
The nonstandard beta PDF is
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WI. —j—r(x — a)a 1(b — x)e 1 for a < x < b B(a, в)(b — a)a+e-1 “ “
(2.96)
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in which a and b are the lower and upper bounds of the beta random variable, respectively; a > 0, в > 0; and B(a, в) is a beta function defined as
Using the new variable Y = (X — a)/(b — a), the nonstandard beta PDF can be reduced to the standard beta PDF as
f B(y | а, в) = 1 Уа—1(1 — y)e—1 for 0 < y < 1 (2.98)
B(а, в)
The beta distribution is also a very versatile distribution that can have many shapes, as shown in Fig. 2.23. The mean and variance of the standard beta random variable Y, respectively, are
а ав
11У = а + в °У = (а + в + 1)(а + в )2 ‘
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When а = в = 1, the beta distribution reduces to a uniform distribution as
