Homework Solutions
Utah State University
ECE 6010
Stochastic Processes
Homework # 7 Solutions

Suppose
is a homogeneous Poisson process
with parameter
. Define a random variable
as the
time of the first occurrence of an event. Find the p.d.f. and the
mean of
.
p.d.f. :
We have,
and , i.e. .Therefore,
Mean :

Suppose
is a w.s.s. random
process with autocorrelation function
. Show that if
is continuous at
then it is continuous for all
.
(CauchySchwartz)
So, 
Under the conditions of problem 2, show that for
,
Let . is nonnegative, nondecreasing on and symmetric about 0. Then,

Suppose
and
are random variables with
and
. Define the random processes
and
by
Mean :
Autocorrelation :
Similarly,
Cross correlation :

Homogeneous Poisson
Simply take and substitute it back into the differential equation and show that it works.

Inhomogeneous Poisson
Simply take and substitute it back into the differential equation and show that it works.
Covariance: Assume that :

Suppose
is a random 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 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.