Homework Solutions
Utah State University
ECE 6010
Stochastic Processes
Homework # 6 Solutions

Suppose
is a sequence of independent
r.v.s each of which is uniformly distributed on the interval
. Define a sequence of r.v.s
by
,
where
. Show that
converges in distribution to an exponential
r.v. with p.d.f.
Here,

Suppose
(i.p.) and that there is a constant
such that
for all
. Show that
(m.s.)
We have (i.p.) and . Therefore,
Define,
and
Taking limit we have . Therefore we have,Therefore, (i.p.) (m.s.) if .

Suppose
(in distribution), where
is a
constant. Show that
(i.p.)
(i.p.) .
(by convergence in distribution)
Therefore,