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)