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# Homework Assignments

## Homework 7

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 .
2. Suppose { is a w.s.s. random process with autocorrelation function . Show that if is continuous at then it is continuous for all . (Hint: Use the Schwartz inequality.)
3. Under the conditions of problem 2, show that for ,

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 .
5. Let , where is a homoegenous Poisson counting process with rate . Show that the differential equation

is solved by

6. Let , where is an inhomoegenous Poisson counting process with time-varying rate , where is a continuous nonnegative function. Show that the differential equation

is solved by

Also, determine the mean and autocorrelation functions of .

7. Suppose is a random process with power spectral density

Find the autocorrelation function of .
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 .

9. Suppose that is a w.s.s., zero-mean, Gaussian random process with autocorrelation function and power spectral density . Define the random process by . Find the mean, autocorrelation, and power spectral density of .
10. Suppose and are independent random variables with and . Define random processes by

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