Gamma Distribution - Section 4.6
The Gamma Distribution is used to model continuous data whose values are positive and have a
probability histogram that is skewed to the right. The properties of the gamma distribution are
•The parameters for the gamma distribution are α > 0 and β > 0. The αparameter is
referred to as the shape parameter and the βparameter is referred to as the scale parameter
for reasons will be investigating.
•The probability density function for the gamma distribution is
f(y) = yα−1e−y/β
βαΓ(α)0≤y < ∞
where Γ(α) = R∞
0yα−1e−ydy.
•In most circumstances, the distribution function F(y) does not have a closed form solution.
Therefore, the probabilities for the gamma distribution must be obtained from tables or from
a statistical software package.
Here are graphs of the probability density function and the distribution function of a Gamma
distribution with α= 2 and β= 2.
p.d.f. of Gamma Distribution with alpha = 2 and beta = 2
y
f(y)
0 5 10 15
0.0 0.05 0.10 0.15
Distribution of Gamma Distribution with alpha = 2, beta = 2
y
F(y)
0 5 10 15
0.0 0.2 0.4 0.6 0.8 1.0
•The theoretical mean of the gamma distribution is
µ=E(Y) = αβ
•The variance of the gamma distribution is
σ2=V(Y) = αβ2
Working with the gamma distribution in R.
To find the probability P(Y≤y) the command in R is
pgamma(y, shape = alpha, scale = beta)
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