Exercise 2.1 (a) Find the Erlang density f S n ( t ) by convolving f X ( x ) = λ exp(− λx ) with itself n times. (b)   Find the moment generating function (MGF) of X (or find the Laplace transform of...

Exercise 2.1 (a)
Find the Erlang density fS
n
(t) by convolving fX
(x) =
λ
exp(−λx) with itself
n
times.

(b)
Find the moment generating function (MGF) of
X
(or find the Laplace transform of fX
(x)), and use this to find the moment generating function (or Laplace transform) of
S
n
=
X1 +
X2 + ··· +
X
n. Invert your result to find fS
n
(t).

(c)
Find the Erlang density by starting with (2.15) and then calculating the marginal density for
S
n.