The spreading of a random variable over its range is measured by the variance, which is defined for the continuous case as
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(X — fZx)2 fx(x) dx (2.36)
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The variance is the second-order central moment. The positive square root of the variance is called the standard deviation ax, which is often used as a measure of the degree of uncertainty associated with a random variable.
The standard deviation has the same units as the random variable. To compare the degree of uncertainty of two random variables with different units, a dimensionless measure ^x = ax/^x, called the coefficient of variation, is useful. By its definition, the coefficient of variation indicates the variation of a random variable relative to its mean. Similar to the standard deviation, the second- order L-moment Л2 is a measure of dispersion of a random variable. The ratio of Л2 to Л1, that is, t2 = k2/k1, is called the L-coefficient of variation.
Three important properties of the variance are
1. Var(a) = 0 when a is a constant. (2.37)
2. Var(X) = E(X2) — E2(X) = ^ — ii2x (2.38)
3. The variance of the sum of several independent random variables equal the sum of variance of the individual random variables, that is,
Var f Y akXЛ = Y a (2.39)
k = 1 ) k = 1
where ak is a constant, and ak is the standard deviation of random variable Xk, k = 1,2,…, K.
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Example 2.11 (modified from Mays and Tung, 1992) Consider the mass balance of a surface reservoir over a 1-month period. The end-of-month storage S can be computed as
in which the subscript m is an indicator for month, Sm is the initial storage volume in the reservoir, Pm is the precipitation amount on the reservoir surface, Im is the surface-runoff inflow, Em is the total monthly evaporation amount from the reservoir surface, and Tm is the controlled monthly release volume from the reservoir.
It is assumed that at the beginning of the month, the initial storage volume and total monthly release are known. The monthly total precipitation amount, surface-runoff inflow, and evaporation are uncertain and are assumed to be independent random variables. The means and standard deviations of Pm, Im, and Em from historical data for month m are estimated as
E (Pm) = 1000 m3, E (Im) = 8000 m3, E (Em) = 3000 m3 a (Pm) = 500 m3, a (Im) = 2000 m3, a (Em) = 1000 m3
Determine the mean and standard deviation of the storage volume in the reservoir by the end of the month if the initial storage volume is 20,000 m3 and the designated release for the month is 10,000 m3.
Solution From Eq. (2.31), the mean of the end-of-month storage volume in the reservoir can be determined as
E ( Sm+1) = Sm + E ( Pm) + E (Im) — E ( Em) — Tm
= 20, 000 + 1000 + 8000 — 3000 — 10, 000 = 16, 000 m3
Since the random hydrologic variables are statistically independent, the variance of the end-of-month storage volume in the reservoir can be obtained, from Eq. (2.39), as
Var( Sm+1) = Var( Pm) + Var( Im) + Var( Em)
= [(0.5)2 + (2)2 + (1)2] x (1000m3)2 = 5.25 x (1000m3)2
The standard deviation and coefficient of variation of Sm+1 then are
a(Sm+1) = V525 x 1000 = 2290m3 and Q(Sm+1) = 2290/16,000 = 0.143
