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MATH6038: Week 4*

February 25, 2016 in MATH6038 | 2 comments

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

 

Week 4*

We introduced matrices as linear maps and discussed some of their properties..

Week 5*

A Maple week first of all. See below for information on lab split.

We will begin talking about the Gauss-Jordan Algorithm for finding matrix inverses  and see how they are useful in solving matrix equations. In particular, we will learn that we can rewrite and solve a linear system as a matrix equation.

Read the rest of this entry »

MATH6038: Week 3*

February 18, 2016 in MATH6038 | Leave a comment

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

I have sent an email about indicating your availability for doing Maple at 18:00 and what Wednesday during Easter doesn’t suit you. 

I also sent an email to some of ye regarding manuals.

Week 3*

After introductions, we looked at systems of linear equations and introduced the idea of writing them in augmented matrix form and simplifying using Gaussian Elimination. We will discuss how to determine if the linear system has a unique, infinite or no solution.

Week 4*

We will introduce matrices as linear maps and discuss some of their properties.

Read the rest of this entry »

MATH6038: Week 2*

February 11, 2016 in MATH6038 | Leave a comment

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

I have sent an email about indicating your availability for doing Maple at 18:00 and what Wednesday during Easter doesn’t suit you. 

I also sent an email to some of ye regarding manuals.

Week 2*

After introductions, we looked at systems of linear equations and introduced the idea of writing them in augmented matrix form and simplifying using Gaussian Elimination. We will discuss how to determine if the linear system has a unique, infinite or no solution.

Week 3*

We will probably just do more examples of solving linear systems.

Read the rest of this entry »

MATH6038: Week 12

May 1, 2014 in MATH6038 | 2 comments

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

Linear Algebra — Test Solutions

More comments on Linear Algebra next week but until then here are the solutions to the test.

Week 12

We spoke about sampling in more detail and also introduced control charts.

Week 13

We will hold a review class on Wednesday 7 May in the usual room. First off, the layout of your exam is the same as Autumn 2013: do question one worth 50/100 and two out of questions two, three, four; each worth 25/100.

I will field any questions ye might have at this time and if there are no questions we will do this exam paper. The best possible thing for your study is to do this exam paper and then on Wednesday see how you got on.

Formulae to Learn?

I spoke about asking Dr. Michael Brennan about putting the confidence interval formula on the exam paper… I just realised this is moot as the exams have already gone for printing. Therefore there are two formulae you will need to know.

The first is the confidence interval for small samples (here I give three presentations):

\bar{x}\pm t\frac{\sigma}{\sqrt{n}}

\bar{x}-t\frac{\sigma}{\sqrt{n}}\leq \mu\leq \bar{x}+t\frac{\sigma}{\sqrt{n}}

\left[\bar{x}-t\frac{\sigma}{\sqrt{n}},\bar{x}+t\frac{\sigma}{\sqrt{n}}\right]

Here \bar{x} is the sample mean, n is the sample size, t is a number of standard deviations, read from a table, depending on n-1 and the level of uncertainty. \sigma is the population standard deviation which we approximate using the sample standard deviation, \sigma\approx s. Finally \mu is the population mean, the number we are trying to get a handle on.

The second set of formulae are like confidence intervals for the control charts aka the inner and outer control limits (ICL & OCL). These should be considered versions of the above formula and I will display them as such… the A_i depends on the level of uncertainty and the sample size, and plays the role of \frac{t}{\sqrt{n}}, while the mean range, \bar{w}, is a measure of the spread\deviation of the data, and plays the role of \sigma. The role of \bar{x} is played by \bar{\bar{x}} — the grand mean:

ICL: \bar{\bar{x}}\pm A_{0.025}\bar{w},

OCL: \bar{\bar{x}}\pm A_{0.001}\bar{w}

ICL: \bar{\bar{x}}- A_{0.025}\bar{w}\leq \bar{x}\leq \bar{\bar{x}}+ A_{0.025}\bar{w},

OCL: \bar{\bar{x}}- A_{0.001}\bar{w}\leq \bar{x}\leq \bar{\bar{x}}+ A_{0.001}\bar{w}

ICL: [\bar{\bar{x}}- A_{0.025}\bar{w}, \bar{\bar{x}}+ A_{0.025}\bar{w}],

OCL: [\bar{\bar{x}}- A_{0.001}\bar{w}, \bar{\bar{x}}+ A_{0.001}\bar{w}]

Here \bar{\bar{x}} is the grand mean — that is the mean of all the data — which may conveniently be calculated as the mean of the sample means. The A_{i} are numbers read from a table, depending on the sample size n. \bar{w} is the mean range — the mean of the sample ranges. Finally \bar{x} is a sample mean.

The smaller numbers are the lower limits and the larger numbers are the upper limits.

Independent Learning: Exercices

You are supposed to be working outside of class and I am supposed to help you with this. Working outside of class means doing the exercises in the notes. Any work that is handed up will be corrected by me. Also you can ask me a question here on this site and I will answer it ASAP.

Questions that you can do at this point include:

  • P. 130 Q. 1-13
  • P.139, Q. 1-7, particularly Q. 1

I wouldn’t necessarily advise looking at ALL of these questions: I would focus on the exam questions over the other questions.

Academic Learning Centre

Those in danger of failing need to use the Academic Learning Centre. As you can see from the timetable there is evening support. You will get best results if you come to the helpers there with specific questions.

Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly.

Additional Notes: E-Books

If you look in the module descriptor, you will see there is some suggested reading. Of course I think my notes are perfect but if you can look here, search for ‘Bird Higher Engineering’ you will see that the library have an E-Book resource.

MATH6038: Week 11

April 14, 2014 in MATH6038 | 6 comments

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

Week 11

We looked at the normal distribution in more detail and talked about sampling.

Week 12

We will talk about sampling in more details and also introduce control charts.

Independent Learning: Exercices

You are supposed to be working outside of class and I am supposed to help you with this. Working outside of class means doing the exercises in the notes. Any work that is handed up will be corrected by me. Also you can ask me a question here on this site and I will answer it ASAP.

Questions that you can do at this point include:

  • P. 125 Q. 1-21, particularly Q. 5-10

I wouldn’t necessarily advise looking at ALL of these questions: I would focus on the exam questions over the other questions.

Academic Learning Centre

Those in danger of failing need to use the Academic Learning Centre. As you can see from the timetable there is evening support. You will get best results if you come to the helpers there with specific questions.

Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly.

Additional Notes: E-Books

If you look in the module descriptor, you will see there is some suggested reading. Of course I think my notes are perfect but if you can look here, search for ‘Bird Higher Engineering’ you will see that the library have an E-Book resource.

MATH6038: Week 10

April 3, 2014 in MATH6038 | 2 comments

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

Maple Catch Up

If you want to do Maple catch up, if for example you miss a lab, please download the Maple file from your email, do the exercises in Maple (either in a new worksheet or on top of the original file with the exercises), save the worksheet and email me the worksheet you were working on. Please complete your Maple catch up before 30 April.

Summer 2011 Q. 2 (b)

This question came up last night however I said that there was an issue with interpretation but that I felt there was only one reasonable interpretation: that the question required a mix of the Poisson Distribution and the Binomial Distribution, and I have answered the question under that assumption below.

I stated however that you will not get a question like this in your exam because your exam will prompt you to use the Binomial, Poisson or indeed Normal distribution.

I have no idea where I got this question from… it wasn’t on any MATH6038 paper… a mystery!

A production department has 35 milling machines. The number of breakdowns on each machine averages 0.06 per week. Determine the probabilities of having

1. one machine breaking down in a week

2. less than three machines breaking down in a week

Solution: 

1. With an average rate of occurrence, \lambda=0.06 given, assume that the number of times a machine breaks down in a week, X, has a Poisson distribution, X\sim \text{Poi}[0.06]. Therefore to calculate the probability that a machine breaks down during the week we look at the probability of the machine breaking down one or more times:

\mathbb{P}[X\geq 1]=1-\mathbb{P}[\text{not-}(X=0)]=1-e^{-0.06}\approx 0.05824.

2. Now we are looking at 35 ‘trials’ where success is defined as a machine breaking down so that we have p=1-e^{-0.06} — and 1-p=1-(1-e^{-0.06})=e^{-0.06} (note that using 0.05824 introduces rounding error. Not the end of the world in an exam but perhaps in real life!).

Let Y be the number of successes from the n=35 trials so that Y has a binomial distribution; Y\sim \text{Bi}[25,1-e^{-0.06}]. We are looking at the probability of less than three machines breaking down in a week;

\mathbb{P}[Y<3]=\mathbb{P}[(Y=0)\cup(Y=1)\cup(Y=2)]

=\mathbb{P}[Y=0]+\mathbb{P}[Y=1]+\mathbb{P}[Y=2]

={35\choose 0}(1-e^{-0.06})^0(e^{-0.06})^{35}+{35\choose 1}(1-e^{-0.06})^1(e^{-0.06})^{34}+

{35\choose 2}(1-e^{-0.06})^2(e^{-0.06})^{33}

0.12246+0.26503+0.2786=0.4154.

 

Week 10

We looked at reliability block diagrams, the binomial distribution, the Poisson distribution and we started talking about the normal distribution.

Week 11

We will look at the normal distribution in more detail and talk about sampling.

Independent Learning: Exercices

You are supposed to be working outside of class and I am supposed to help you with this. Working outside of class means doing the exercises in the notes. Any work that is handed up will be corrected by me. Also you can ask me a question here on this site and I will answer it ASAP.

Questions that you can do at this point include:

  • P. 108, Q. 1-7
  • P. 112 Q. 1-17
  • P.116, Q. 1-15

I wouldn’t necessarily advise looking at ALL of these questions: I would focus on the exam questions over the other questions.

Academic Learning Centre

Those in danger of failing need to use the Academic Learning Centre. As you can see from the timetable there is evening support. You will get best results if you come to the helpers there with specific questions.

Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly.

Additional Notes: E-Books

If you look in the module descriptor, you will see there is some suggested reading. Of course I think my notes are perfect but if you can look here, search for ‘Bird Higher Engineering’ you will see that the library have an E-Book resource.

MATH6038: Week 9

March 27, 2014 in MATH6038 | Leave a comment

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

Maple Catch Up

If you want to do Maple catch up, if for example you miss a lab, please download the Maple file from your email, do the exercises in Maple (either in a new worksheet or on top of the original file with the exercises), save the worksheet and email me the worksheet you were working on. Please complete your Maple catch up before 30 April.

Week 9

We did some statistics stuff in Maple and then we will delved further into probability, talking about random variables, independence, conditional probability and tree diagrams.

Week 10

We will look at reliability block diagrams, the binomial distribution, the Poisson distribution and we will start talking about the normal distribution.

Independent Learning: Exercices

You are supposed to be working outside of class and I am supposed to help you with this. Working outside of class means doing the exercises in the notes. Any work that is handed up will be corrected by me. Also you can ask me a question here on this site and I will answer it ASAP.

Questions that you can do at this point include:

  • P.98, Q. 1-11
  • P.103, Q. 1-4

Academic Learning Centre

Those in danger of failing need to use the Academic Learning Centre. As you can see from the timetable there is evening support. You will get best results if you come to the helpers there with specific questions.

Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly.

Additional Notes: E-Books

If you look in the module descriptor, you will see there is some suggested reading. Of course I think my notes are perfect but if you can look here, search for ‘Bird Higher Engineering’ you will see that the library have an E-Book resource.

MATH6038: Week 8

March 20, 2014 in MATH6038 | 3 comments

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

Test

Results pending…

Maple Catch Up

If you want to do Maple catch up, if for example you miss a lab, please download the Maple file from your email, do the exercises in Maple (either in a new worksheet or on top of the original file with the exercises), save the worksheet and email me the worksheet you were working on.

Week 8

We had our test on Linear Algebra first and then after a break we finished off the chapter on statistics and started talking about probability.

Week 9

We will do some statistics stuff in Maple and then we will delve further into probability.

Independent Learning: Exercices

You are supposed to be working outside of class and I am supposed to help you with this. Working outside of class means doing the exercises in the notes. Any work that is handed up will be corrected by me. Also you can ask me a question here on this site and I will answer it ASAP.

Questions that you can do at this point include:

  • P.90, Q. 1-3 (these are particularly important)
  • P.93, Q. 1-10 (these less so)

Maple Online & Wolfram Alpha

If you are subscribed to CIT MathsOnline you will have free access to the mathematical software package Maple:

Self-enrolment for Maths Online

1.           Log into Blackboard Learn

2.           Click on the Courses tab button at the top of the screen. Go to Course Search and type Maths Online in the box.

3.           Once you’ve found the course, click the non-credit-course button and click on Enrol. This should take you to the Self Enrolment page.

4.           Your Access Code is mathsonline (lower case, no spaces).

5.           After you’ve finished click Submit. You should now see a message that says your enrolment was successful.

Once you’ve enrolled, no go back to the Blackboard home page and click on the Maths Online button: it should be under an Academic Learning Support Tab. You can download Maple by selecting the Mathematical Software tab in the left hand column and following the instructions under the Maple item. Click Maple text to start.

If you have any problem with this take a print screen and send it to me and I will try and sort you out.

I myself am not a Maple expert but ‘grew up’ with another mathematical software package Mathematica. Mathematica powers the “computational knowledge engine” WolframAlpha. Go on ask it a question!

Academic Learning Centre

Those in danger of failing need to use the Academic Learning Centre. As you can see from the timetable there is evening support. You will get best results if you come to the helpers there with specific questions.

Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly.

Additional Notes: E-Books

If you look in the module descriptor, you will see there is some suggested reading. Of course I think my notes are perfect but if you can look here, search for ‘Bird Higher Engineering’ you will see that the library have an E-Book resource.

MATH6038: Week 7

March 13, 2014 in MATH6038 | 6 comments

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

Test

LATE EDIT: I just realised that I included no questions like Q.1 on P.32 of the notes… they are also examinable (matrix arithmetic). Also Q. 7 should read “Verify your answer for z using Cramer’s Rule”

The test will be from 7.05-8 pm next week (Wednesday 19 March). I won’t give a sample but instead look here for a selection of test & exam questions. You  can find a summary of Chapter 1 methods here.

Also I explained that some questions involving finding the inverse of 3×3 matrices using the Gauss-Jordan Algorithm can end up very messy in terms of the numbers.  Questions like P. 44 Q.1-5 are fine as are P. 52 Q. 2, 3 (iv)-(vi)

Stay away from questions like the examples 3 & 4 on P. 49 and P.52 Q. 3 (iii), (vii)-(x) and P.62 Q.2, 3. — the numbers here are just too messy.

Maple Catch Up

If you want to do Maple catch up, if for example you miss a lab, please download the Maple file from your email, do the exercises in Maple (either in a new worksheet or on top of the original file with the exercises), save the worksheet and email me the worksheet you were working on.

Week 7

We continued talking about statistics including the standard deviation. We introduced frequency distributions. We skipped over the material that used the assumption of uniformity — we aren’t going to use it.

We introduced the assumed mean method and presented the formulae that are used to calculate the mean and standard deviation of a frequency class distribution using it. We also described the cumulative frequency distribution and how we can use it to find the median.

In Maple we revised for the test with the aid of Maple.

Week 8

We will have our test on Linear Algebra first and then after a break we will finish off the chapter on statistics and start talking about probability.

Independent Learning: Exercices

You are supposed to be working outside of class and I am supposed to help you with this. Working outside of class means doing the exercises in the notes. Any work that is handed up will be corrected by me. Also you can ask me a question here on this site and I will answer it ASAP.

Questions that you can do at this point include:

  • the sample questions for your test
  • p.70 Q. 1-3
  • P.90 Q. 2

Maple Online & Wolfram Alpha

If you are subscribed to CIT MathsOnline you will have free access to the mathematical software package Maple:

Self-enrolment for Maths Online

1.           Log into Blackboard Learn

2.           Click on the Courses tab button at the top of the screen. Go to Course Search and type Maths Online in the box.

3.           Once you’ve found the course, click the non-credit-course button and click on Enrol. This should take you to the Self Enrolment page.

4.           Your Access Code is mathsonline (lower case, no spaces).

5.           After you’ve finished click Submit. You should now see a message that says your enrolment was successful.

Once you’ve enrolled, no go back to the Blackboard home page and click on the Maths Online button: it should be under an Academic Learning Support Tab. You can download Maple by selecting the Mathematical Software tab in the left hand column and following the instructions under the Maple item. Click Maple text to start.

If you have any problem with this take a print screen and send it to me and I will try and sort you out.

I myself am not a Maple expert but ‘grew up’ with another mathematical software package Mathematica. Mathematica powers the “computational knowledge engine” WolframAlpha. Go on ask it a question!

Academic Learning Centre

Those in danger of failing need to use the Academic Learning Centre. As you can see from the timetable there is evening support. You will get best results if you come to the helpers there with specific questions.

Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly. For example this question explaining why we have different formulae for the population and sample standard deviation.

Additional Notes: E-Books

If you look in the module descriptor, you will see there is some suggested reading. Of course I think my notes are perfect but if you can look here, search for ‘Bird Higher Engineering’ you will see that the library have an E-Book resource.

MATH6038: Week 6

March 7, 2014 in MATH6038 | Leave a comment

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

Test

LATE EDIT: I just realised that I included no questions like Q.1 on P.32 of the notes… they are also examinable (matrix arithmetic). Also Q. 7 should read “Verify your answer for z using Cramer’s Rule”

The test will be from 7.05-8 pm in Week 8 (Wednesday 19 March). I won’t give a sample but instead look here for a selection of test & exam questions. You  can find a summary of Chapter 1 methods here.

Also I explained that some questions involving finding the inverse of 3×3 matrices using the Gauss-Jordan Algorithm can end up very messy in terms of the numbers.  Questions like P. 44 Q.1-5 are fine as are P. 52 Q. 2, 3 (iv)-(vi)

Stay away from questions like the examples 3 & 4 on P. 49 and P.52 Q. 3 (iii), (vii)-(x) and P.62 Q.2, 3. — the numbers here are just too messy.

Maple Catch Up

If you want to do Maple catch up, if for example you miss a lab, please download the Maple file from your email, do the exercises in Maple (either in a new worksheet or on top of the original file with the exercises), save the worksheet and email me the worksheet you were working on.

Week 6

We solved a number of linear systems using the  inverse matrix method. Note however that this cannot work if the solution is not unique… we used a property of matrices called the determinant to decide in advance if a certain class of linear system, homogenous systems, had a unique solution or not. If the matrix (of coefficient) is invertible then the solution has to be unique and we saw that the matrix (of coefficients) is invertible if and only if the determinant is non-zero. Finally we will looked at Cramer’s Rule: a method for solving for only one of the variables of a linear system. We then began talking about statistics and got as far as discussing the difference between the mean, the median and the mode.

Week 7

We will go as far into the chapter on statistics as possible. In Maple I will allow you to revise for the test with the aid of Maple. I will write a worksheet which includes the sample questions. Remember Maple runs from 6-7.10 and 8.50-10. The theory class runs from 7.15-8.45.

Independent Learning: Exerices

You are supposed to be working outside of class and I am supposed to help you with this. Working outside of class means doing the exercises in the notes. Any work that is handed up will be corrected by me. Also you can ask me a question here on this site and I will answer it ASAP.

Questions that you can do at this point include:

  • the sample questions
  • p.52 Q. 3 (iv)-(vi)
  • P.62 Q. 1 and 4
  • P. 68 Q. 1-4

Maple Online & Wolfram Alpha

If you are subscribed to CIT MathsOnline you will have free access to the mathematical software package Maple:

Self-enrolment for Maths Online

1.           Log into Blackboard Learn

2.           Click on the Courses tab button at the top of the screen. Go to Course Search and type Maths Online in the box.

3.           Once you’ve found the course, click the non-credit-course button and click on Enrol. This should take you to the Self Enrolment page.

4.           Your Access Code is mathsonline (lower case, no spaces).

5.           After you’ve finished click Submit. You should now see a message that says your enrolment was successful.

Once you’ve enrolled, no go back to the Blackboard home page and click on the Maths Online button: it should be under an Academic Learning Support Tab. You can download Maple by selecting the Mathematical Software tab in the left hand column and following the instructions under the Maple item. Click Maple text to start.

If you have any problem with this take a print screen and send it to me and I will try and sort you out.

I myself am not a Maple expert but ‘grew up’ with another mathematical software package Mathematica. Mathematica powers the “computational knowledge engine” WolframAlpha. Go on ask it a question!

Academic Learning Centre

Those in danger of failing need to use the Academic Learning Centre. As you can see from the timetable there is evening support. You will get best results if you come to the helpers there with specific questions.

Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly. For example this question about the determinant.

Additional Notes: E-Books

If you look in the module descriptor, you will see there is some suggested reading. Of course I think my notes are perfect but if you can look here, search for ‘Bird Higher Engineering’ you will see that the library have an E-Book resource.