ECE2191

Probability Models in Eng.

Example 3

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 question 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

Assume that two sources A and B are producing noise,where the power of the noise from each source is modelled by a Gaussian random variable; with mean of 230 milliwatts and standard deviation of 5 milliwatts for source A and mean of 225 milliwatts and standard deviation of 10 milliwatt for source B. Assume that the probability of the noise power being negative is negligible.Which source is more likely to produce a noise stronger than 240 milliwatts.

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.)

The following information may be used:

Q-function values:

Question 2

The number of new babies born with a congenital disease over time is modelled by a Poisson random variable,with an average of 1460 new cases per year. Assume that no leap year is considered.

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.)

2a) (a)What is the probability of no babies born with the disease in a day?

2b) (b)What is the probability of no more than 2 babies born with the disease in a day?

Section B-Hand-Written/Drawn Response Questions

Question 3

ECE2191 is offered in both Clayton and Malaysia,with 60 male and 40 female students enrolled in Clayton and 40 male and 10 female students enrolled in Malaysia. A group of 8 students are randomly selected as ECE2191 student representatives.

a) How many ways are there to choose this group?(Calculation of the final answer is not required.)

b) What is the probability that a randomly selected group has exactly 4 female and 4 male students?(Calculation of the final answer is not required.)

c) What is the probability that a randomly selected group has exactly 5 male students from Clayton, 1 female student from Clayton, 1 male student from Malaysia and 1 female student from Malaysia? (Calculation of the final answer is not required.)

Provide answers to all parts (a) to (c). Explain your results in full 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.)

Question 4

The PDF of a continuous random variable Xis modelled as:

a) If the expected value is 0.6,find the constantsa and b.

b) Find an expression for the CDF.

c) Calculate the variance.

d) Find P(X>0.5).

e) Find the variance of the new random variable Y=2X+1.

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,Y) be a pair of random variables with joint PMF given by:

Answer the following two questions:

(a) Are X and Y independent?

(b) Are X and Y uncorrelated?

Provide answers to both questions (a) and (b). Justify your answers 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 6

You're at a large dinner party and the host has hired an up-and-coming chef to create his famous seafood dish for the occasion.Given the intricacy of the dish,each serving of the dish must be individually prepared.The amount of time required to prepare a single serving of the dish is normally distributed with a mean of 45 minutes and a standard deviation of 10 minutes. A random sample of 16 servings is selected.

Answer the following two questions:

(a) What is the probability that the sample mean is between 45 and 52 minutes?

(b) 95%of allsample means will fall between which two values?

Provide answers to both questions (a) and (b). Justify your answers 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

You run a local restaurant that sells noodles.Your restaurant is part of a major franchise consisting of noodles  restaurants all over the world.You are  interested in studying diners'patterns of behaviour. In particular, you want to find an estimate of the average time that diners spend in the franchise's noodles restaurants all over the world. You know  that the population standard deviation (i.e., the  standard deviation of the time that diners spend in the franchise's noodles restaurants all over the world) is 15; and you can safely assume that the times diners spend in the franchise's noodles restaurants all over the world follow a normal distribution.You collect a sample of your own local diners' times spent in your restaurant during an evening of food service. These lengths of time are provided below (in minutes):

32,40,33,57,16,54,29,23,73,40,46,71,48,53,54,10,44,24,73,32

a) Construct a 99% confidence interval estimate for the average time that diners spend in the franchise's noodles restaurants all over the world.

b) Additionally,construct a 90% confidence interval estimate for the time that diners spend in the franchise's noodles restaurants allover the world, and discuss this 90% 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.)