Cumulative distribution function
The cumulative distribution function of the Gamma distribution is
\[ F(x) \;\;=\;\; P(X \le x) \;\;=\;\; \int_0^x {\frac {\beta^\alpha }{\Gamma(\alpha)} u^{\alpha - 1} e^{-u\beta}} \;du \]This integral cannot be simplified and can only be evaluated numerically. In Excel, the following function can be used.
= GAMMA.DIST( \(x\), \(\alpha\), \(\beta\), true)
Question
If a random variable, \(X\), has a Gamma distribution
\[ X \;\;\sim\;\; \GammaDistn(\alpha = 7,\; \beta = 12) \]what is the probability of getting a value between 0.5 and 1.0?
(Solved in full version)
Quantiles from Gamma distributions
In a similar way, there is no algebraic formula for the quantiles of a Gamma distribution, but computer algorithms are available to find them numerically. To find the value \(x\) such that \(F(x) = q\), the following Excel function can be used.
= GAMMA.INV( \(q\), \(\alpha\), 1/\(\beta\))
Question
If a random variable, \(X \sim \GammaDistn(\alpha = 7,\; \beta = 12)\), what is the lower quartile of its distribution?
(Solved in full version)