Homework Solutions
Utah State University
ECE 6010
Stochastic Processes
Homework # 9 Solutions

Suppose
is a ramdom process with
power spectral density
Similarly,

Suppose that
is a random variable with
p.d.f.
and
is a random variable independent
of
uniformly distributed in
. Define a random
process by
where
is a constant. Find the power spectral
density of
.
Therefore, 
Suppose events occur randomly in
in
the following way:
 The numbers of events in nonoverlapping intervals are independent of one another.
 exactly one event in , where is a continuous nonnegative function on .
 more than one event in an interval of length

For
, show that
is a
Poisson random variable with parameter
.
k events in for and .
If satisfies the above equation then we can say that is a Poisson random variable with parameter .
Now, let and
Therefore,
So that does satisfy. And is Poisson.

Find the mean and autocorrelation functions of
.
We have, . So,
Now,
So,

Suppose that
is a w.s.s.,
zeromean, Gaussian random process with autocorrelation function
and power spectral density
. Define the random process
by
. find the mean, autocorrelation, and powerspectral
density of
.
Mean :
PSD :

Suppose
and
are independent random variables
with
and
. Define random processes by
Similarly,
So, and are individually WSS, but not jointly WSS.