Subject: Engineering   / General Engineering
Question
1. Reference example 9.3 on page 464 of the text.
This example shows that the ML estimator of the customer arrival rate is
ˆ n
??
Yn
Recall that the arrival time of the ith customer was designated Yi , which was assumed distributed
Poisson, and the inter-arrival times, X i , were distributed exponential;
X i ~ f X ? x ? ? ? e ?? x
Note that ? has the dimensions of rate (1/time) and that x has the dimensions of time.
a) Show the PMF for the arrival times in terms of the estimated arrival rate, ? . Keep in mind that
? represents an arrival rate, and the Poisson PMF describes the probability of the number of
arrivals in a time interval.
b) Describe in words how you would implement this ML estimator. In other words, describe the
actual procedure for estimating ? .
2. You are measuring a voltage that has an additive noise component. An appropriate model is
X ?Y ? N
where N is the noise, which is distributed N ? 0, ? ? .
a) Derive the ML estimate of the signal, V.
b) Derive the ML estimate of the noise variance.
Note that the samples are given by
xi ? V ? ni
and that you may assume the xi to be I.I.D.
3. a) Derive the 90% confidence interval for the signal, V, in problem 2 based on n samples,
assuming that the noise variance is known.
b) Repeat for an approximate 90% confidence interval in the case of unknown variance (which
you must estimate).
4. Write a Matlab script to analyze the data of example 9.9, page 477 in the text. Plot the data,
calculate the linear regression fit, and plot the fit along with the data.
