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

1. 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,

So,
and , i.e. .

Therefore,

Mean :

2. Suppose is a w.s.s. random process with autocorrelation function . Show that if is continuous at then it is continuous for all .

 (Cauchy-Schwartz)

So,

So continuous at all .

3. Under the conditions of problem 2, show that for ,

Let . is non-negative, non-decreasing on and symmetric about 0. Then,

Now, our example corresponds to:

4. Suppose and are random variables with and . Define the random processes and by

Find the mean, autocorrelation, and cross correlations of these random processes in terms of the moments of and .

Mean :

Autocorrelation :

Similarly,

Cross correlation :

5. Homogeneous Poisson

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

6. Inhomogeneous Poisson

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

Mean:

Covariance: Assume that :

Similarly when . Then

7. Suppose is a random process with power spectral density

Find the autocorrelation function of .

Similarly,

Therefore,

8. 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,

9. Suppose that is a w.s.s., zero-mean, Gaussian random process with auto-correlation function and power spectral density . Define the random process by . find the mean, autocorrelation, and powerspectral density of .

Mean :

Autocorrelation :

PSD :

10. Suppose and are independent random variables with and . Define random processes by

Find the autocorrelation and cross-correlation functions of and . Are and jointly wide sense stationary? Are they individually wide sense stationary?

Similarly,

So, and are individually WSS, but not jointly WSS.