EE5251 PROBLEM SET #2 Solution

Problem 1
Do problem 2.5 in your text.
Problem 2
Do problem 2.10 in your text. In addition do the following:
h) Are X and Y independent? Show this by determining the marginal distributions
fX(x) and fy(y).
Problem 3
Number of Residents Number of Residents with
No Felony Conviction One or More Felony Convictions

Female Residents 359,881 172,742 Male Residents 340,454 40,854
You are visiting an island that is a prison colony for political prisoners from the dystopian planet Tralfamador. When a political prisoner receives a guilty verdict in court, (s)he is declared a felon and sent to the island along with their immediate family members. The felony conviction statistics for the current residents of the island are given the table above. Consider the following outcomes for a resident of the island picked at random: (1) M = male resident;(2) F = female resident; (3) C = convicted felon; (4) I = no felony record. Using the data given and assuming that relative frequencies are equal to probabilities, calculate the following: P(M), P(F), P(C), P(I), P(M ∩I), P(F ∩I), P(M ∩C), P(F ∩C). Now using Bayes’ rule answer the following questions. Be sure to check your answers using relative frequencies computed from the given data.
a) What is the probability that a male resident of the island picked at random will be a convicted felon?
b) What is the probability that a female resident of the island picked at random will be a convicted felon?
c) What is the probability that an island resident picked at random is female, given that that they are a convicted felon?
d) What is the probability that an island resident picked at random is male, given that they are a convicted felon?
Problem 4
Do problem 3.9 in your text. This is the problem we kicked-off the course with and now you have the tools to find its solution. HINT: Treat it as a recursive least squares problem.
Problem 5

Figure 1: A system with an unknown mathematical model.
This problem demonstrates the utility and power of a simple, least squares estimator. You will use least squares to determine the mathematical model of an unknown dynamical system. The system can be an unknown electrical circuit, for example, which you can stimulate and then record its output (e.g., voltage-in-voltage out). Or alternately, it can be a vehicle whose equations of motion your do not know. So your apply control inputs (e.g., turn the vehicle right and left rapidly; accelerate and decelerate it, etc) and observe its kinematic state (e.g., position, velocity, acceleration, etc). Such problems are called system identification (or simply sysid) and they are a small part of what we know as machine learning today.
Download the data file inoutdata.txt found on the assignment page. This data contains the output from the unknown (or “blackbox”) system shown in Figure 1. The first column of the data is the time tk, the second column is the input uk and the third column is the output yk (that is, yk when vk = 0). The system dynamics and observation models are:
e
αk = Fαk−1 + Guk−1 yk = y + vk = Cαk + vk e
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Suppose you know that the matrices F, G and H have the following form:

Your job is to use least squares to estimate (or identify) the parameters or entries of F and G. That is, generate an optimal estimate of:

You will proceed to do this as follows:
a) Write out the measurement equation for your estimator in the form
y = Hx e
where y is the error-free sensor measurement (i.e., case when v = 0). Clearly identifying e
the entries in y and H. This is by far the most challenging part of the problem. HINT: e
Try to relate yk to yk−1,yk−2,...,uk−1,uk−2,... and so using an autoregrsive moving average (ARMA) model. If you do not know what an ARMA model is, look it up.
b) Estimate the state vector using normal least squares.
c) Note that error-free is not available to you in practice. Instead you will use some sensor to measure an error corrupted version denoted yk. Use the MATLAB command randn to simulate the sensing process as follows:
yk = yk + 0.1 ∗randn(N,1) e
where N is the number of data points in yk you are using for the estimation process. e
d) Use the history of you generated and uk to generate an estimate of x.
e) Repeat the estimation process but now use a less noisy sensor whose output yk is given by:
yk = yk + +0.001 ∗randn(N,1) e Does the value of x change appreciably?
f) Comment on how you would validate the accuracy of the estimate you generated.
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