ECE2191

Probability Models in Eng.

Example 2

Information

This is an open-book exam.You are permitted to use lecture notes and calculators to complete the questions.

This exam consists of 7 questions.

You have 2 hours and 10 minutes to complete the exam.

The exam has the following sections:

· Section A: Essay and Composite Questions

o Answer the question in section A by entering text online in the dedicated space below each question.Note that these questions may have multiple parts and all parts should be attempted.

· Section B: Hand-written/drawn response Questions

o Answer all of the hand-written/drawn response questions in section B on your own pieces of paper.All questions in section B should be attempted.Please clearly label each blank piece of paper with your Student ID and the question number(and subpart of the question,if applicable). Please do not write your name on the paper.You will have time at the end of your exam to upload photographs of your answer sheets.

If you believe there is an error or a question is unclear,clearly state any assumptions you need to make and proceed.

Section A: Essay and Composite Questions

Question 1

We collect digital output samples from a sensor and note that some samples may be corrupted by random noise(independently).Assume that the number of noise samples that we collect over time is modelled by a Poisson random variable, with an average of 600 noise samples per minute.

Answer the following questions.Explain your results in fullby including all reasoning, formulae used and/or calculations performed to arrive at your answers. (Note: no marks will be given if no reasoning, formulae used and/or calculations are provided.)

1a) a)What is probability that no noise sample is collected in a 200-ms period?(3 marks)

1b) b)What is the probability of collecting no more than 2 noise samples in a 200-ms period?(3 marks)

Question 2

A biomedical engineer should choose between two sensors for use in a medical device.The lifetime of each sensor is modelled by a Gaussian random variable,where for Sensor 1,the mean is 28,000 hours  and standard deviation is 1000 hours and for Sensor 2,the mean is 24,000 hours and standard deviation is 4000 hours.Assume that the probability of the lifetime being negative is negligible.Which sensor should be chosen if the desired lifetime of the device is 30,000 hours?Include all justification,  reasoning,formulae used(in a text format)and/or calculations performed to arrive at your answer.

(Note:no marks will be given if no reasoning,formulae used and/or calculations are provided.)

The following information may be used:

Q-function values:

Section B: Hand-written/drawn response Questions

Question 3

The body temperature values of a group of patients were collected.If we model the body temperature Xin Celsius as a random variable with probability density function

a) Calculate the constanta.

b) Calculate the cumulative distribution function.

c) What is the probability that the body temperature of the patient is  or higher?

d) What is the expected value of the body temperature?

e) Knowing that  explain how the variance of the body temperature in Fahrenheit is related to the variance of the body temperature in Celsius.

Explain your results in full by including all reasoning,formulae used and/or calculations performed to arrive at your answers.(Note:no marks will be given if no reasoning,formulae used and/or calculations are provided.)

Question 4

It is the year 2025, and ECE2191 is offered to students all over the world.As a tutor of ECE2191 at a local university campus, your job this semester has been to grade the assessments of a smallcohort of local students that Monash University allocated to you.Given that the semester and exam periods are now over, you are interested in conducting some analysis regarding students' final marks. In particular,you want to find an estimate of the average final mark of ECE2191 students from all over the globe. You know that the population standard deviation (i.e., the standard deviation of the final mark of ECE2191 students from all over the world) is 10; and you can safely assume that the final marks of ECE2191 students from all over the world follow a normal distribution. Your own local students' final marks are provided below (marks are out of 100):

56, 67, 40, 62, 70, 62, 68, 67, 52, 53, 79, 60, 56, 77, 78, 70, 53, 59, 68, 64, 63, 58, 49, 66, 54, 75, 66, 64, 75, 88

a) Construct a 99%confidence interval estimate for the average final mark of ECE2191 students from all over the world.

b) Construct a 95%confidence interval estimate for the average final mark of ECE2191 students from all over the world, and discuss this 95%confidence interval estimate in comparison to the 99% confidence  interval estimate that you calculated prior.

Explain your results in full by including all reasoning, formulae used and/or calculations performed to arrive at your answers. (Note: no marks will be given if no reasoning, formulae used and/or calculations  are  provided.)

Question 5

Let X be a random variable with probability density function fx (x) and let X₁,….,Xn be a set of independent random variables each with probability density function fx (x). Then the set of random variables X₁ ,...,Xn is called a random sample of size n of X. The sample mean is defined by

Suppose that X has meanμand variance .How many samples of X should be taken if the probability that the sample mean will not deviate from the true mean μby σ/10 or greater is at least 0.95?

Include all reasoning, formulae used and/or calculations performed to arrive at your answer.

(Note: no marks will be given if no reasoning, formulae used and/or calculations are provided.)

Question 6

An engineer has 30 chips of which 11 are broken. Since she does not know which chip is broken, she randomly chooses 5 chips to test.

a) What is the sample space for the number of chosen chips which are faulty?

b) What is the probability that exactly one of the chosen chips is faulty?

c) We know that 5 of the chips are new and 25 of the chips are used (without knowing which ones are  new) and we assume that the probability of a new chip being faulty is 0.2 and the probability of a used chip being faulty is 0.4. If the randomly chosen chip is not faulty, what is the probability that it is new?

Explain your results in full by including all reasoning, formulae used and/or calculations performed to arrive at your answers.(Note: no marks will be given if no reasoning, formulae used and/or calculations are provided.)

Question 7

Consider the binary communication channel shown below in Fig. 1. Let (X,Y) be a pair of random variables, where X is the input to the channel, and Y is the output of the channel. The variable X may assume either the value 0 or the value 1.Similarly, the variable Y may  assume either the value 0 or the value 1. Because of channel noise, an input 0 may convert to an output 1, and vice versa. The channel is characterised by the channel transition probabilities po, qo, P1 and q1,as follows:

Note that 

Your employer asks you the following:

(a) Compute the joint probability mass function for (X,Y).

(b) Compute the marginal  probability mass functions of X and Y.

(c) Are X andY independent?

(d) Can you determine whether there exists a linear relationship between X and Y, and if so, quantify the relative strength of this relationship?

Provide answers to all four of your employer's questions. Justify your answers by including all reasoning, formulae used and/or calculations performed to arrive at your answers. (Note: no marks willbe given if no reasoning, formulae used and/or calculations are provided.)

Fig. 1: Binary communication channel.