## Description

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 yek (that is, yk when vk = 0). The system dynamics and observation models are:

αk = F αk−1 + Guk−1

yk = ye+ vk = Cαk + vk

2

Suppose you know that the matrices F, G and H have the following form:

F =

−a1 1

−a2 0

, G =

b1

b2

, C =

1 0

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:

x =

a1 a2 b1 b2

T

You will proceed to do this as follows:

a) Write out the measurement equation for your estimator in the form

ye = Hx

where ye is the error-free sensor measurement (i.e., case when v = 0). Clearly identifying

the entries in ye and H. This is by far the most challenging part of the problem. HINT:

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.1

b) Estimate the state vector using normal least squares.

c) Note that error-free yek 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 = yek + 0.1 ∗ randn(N, 1)

where N is the number of data points in yek you are using for the estimation process.

d) Use the history of yek 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 = yek + +0.001 ∗ randn(N, 1)

Does the value of x change appreciably?

f) Comment on how you would validate the accuracy of the estimate you generated.

1As of the moment I was writing this homework set (September 5, 2020), the Wikipedia Article on ARMA

is rather verbose and hard to follow. You can read it to learn what an ARMA model is but I would suggest

looking elsewhere to learn about ARMA.

3