so-called complete Gamma
function (
function ). Show that
Solution:
Using integration by parts we have
since the value of the first part is zero. It can easily be proved by the help of the L'Hospital' rule.
It is easy to see that 
, so 
that is 
can be considered as the generalization of the
factorial function.
!Solution:
Introducing the substitution 
, 
thus
Exercise 6.11. Find
the mean, variance and the 
th moment of the gamma distribution with parameters
.
Solution:
where
Introducing the substitution 
since 
.
Similarly
Thus
That is the squared coefficient of variation is 
, that can be less and greater than 
.
Finally
In particular, in the case of 
we have the Erlang distribution with parameters
and we obtain
In case of 
it reduces to the exponential distribution, that
is 
Exercise 6.12. Find
the mean and variance of the Pareto distribution with parameters
.
Solution:
Thus
Exercise 6.13. Let
, and 
, where 
. Find the distribution function of 
.
Solution:
that is we obtain the Pareto distribution with parameters
.