The gamma distribution may be written in several different (but mathematically equivalent) forms. Excel uses the following form for the two-parameter gamma distribution in its functions GAMMADIST (xi) = F(xi) and GAMMAINV(F) = (xi) :

The three-parameter gamma distribution with lower bound is easily evaluated by replacing the lower and upper limits by and respectively. Method-of-moments estimates for the three parameters are
α = 4/C2S
β = Sx ˆ™ CS / 2 = Sx /ˆšÎ±

If CS

The skewness of the two-parameter gamma distribution is
CS = 2 / ˆša= 2CV
Which provides a way of computing frequency factors, as described below.
Of course, moments are computed using logarithms of the data for the LP3 distribution. For given values of α and β, Excel function GAMMADIST returns F(x1) given x1 and GAMMAINV returns xx given F. Letting x1 = + K - Sx where K(F,CS) is the frequency factor, and using the parameter estimates given above, the upper limit of the integral can be manipulated to give
K(F,CS) = Ginv ˆ™ CS / 2-2 / CS,
Where Ginv = GAM MAIN V(F, α, β) To evaluate K, set β = 1 and a = 4 / Cs Hence, frequency factors can be computed only as a function of the CDF and skewness for any values of either parameter, and Table 3-4 can be avoided. For negative skewness, the symmetry of the distribution is exploited, and the same relationship holds for K, but with 1-F as the argument of GAMMAINV, instead of F. (The negative sign of Cs is retained in the equation.) For the special case of Cs = 0, K = z = standard normal variate.
(a) Compute frequency factors for T = 2yr and 100 yr and for CS = + 0.5 and CS = -0.5 Compare the four values with the values given in Table 3-4.
(b) Repeat Problems 3.7, 3.8 and 3.10 using Excel functions GAMMADIST and GAMMAINV.

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