Continuous Random Variable on Gamma Distribution
A
random variable is said to have gamma distribution with parameters (t, λ),
λ > 0, t > 0 if the function of concentration is given by
Gamma
distribution is often found in practice as a distribution of the length of time
a person has to wait until a number of events occur. More specifically, if the
events occur randomly at intervals and fulfill the three axioms it turns out
that the length of time people have to wait until a number of events occur is a
gamma random variable with a parameter (n, λ). The gamma distribution with
parameters (n, λ), n integers, is often called the Erlang -n distribution.
(Note that if n = 1, this distribution is reduced to the exponent
distribution).
Gamma
distribution with λ = 1/2 and t = n / 2 (n posirif integers) is called the
distribution of X2 (read
chi-square). The distribution of chi-square in practice often appears as a
distribution of errors that occur when trying to shoot a target in a dimensionless
space n if every error in the coordinates spreads normally.
Integration
by parts of Γ (t) yields,
For
integer t values, say t = n, by applying the above gamma distribution equation
we will get repeatedly,
Because 
then for values of n integers,
then for values of n integers, 
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