All entries for October 2011

October 30, 2011

Balls in boxes and other miscellaneous thoughts

It is a reasonably common statement in probability that problems can be simplified down to spinners or balls in boxes. So:
Let there be k balls that we wish to place in n boxes. If we can distinguish between the balls, then each ball can go in n different places, so the total is \textstyle n^k. If we cannot distinguish between the balls, this is answered by the occupancy theorem: the multiset formula for n and k.

What, then, if it is the boxes we cannot distinguish between? If the balls also cannot be distinguished between, this is equivalent to partitioning k into at most n parts. The OEIS implies that this has a closed-form solution: \textstyle \frac{1}{n!} \frac{d^n\left( \frac{1}{\prod_{i=1}^k(1-x^i)} \right)}{dx^n}(0), which is nice to know, even if it suggests there’s no ‘nice’ combinatorial solution.
If we can distinguish between the balls but not the boxes, I believe this would come to a sum of Stirling numbers of the second kind: \sum_{i=0}^n \left\{\begin{matrix} k \\ i \end{matrix}\right\}.

Other miscellany: Mario Micallef told us a few integration tricks this week: being able to cancel entire functions while integrating from 0 to 2pi (due to them containing sine), \textstyle\int_0^{2\pi}sin^2(x)dx=\pi by comparison with \cos^2(x) across the same interval. Massively time-saving tricks I’d never considered before. Great!

From combinatorics: wondered if there was a way to check transitivity of a relation expressed in matrix form quickly. Found nothing more than checking each row one step at a time, but I least I can do that fairly quickly now.

October 27, 2011

A4: Memory and Sense

Follow-up to Effective Learning Skills? VARK! from Midgley, Christopher - Pointless twaddle and meaningless diatribes

Tutor was Ceri Marriott.

Take advantage of synaesthesia of memory by associating words with colours/sounds/whatever – give yourself more mental hooks to connect ideas with
This one has so far gone differently to how I expected – I’m remembering things based on the order of introduction and internal links instead of anything else. Haven’t really noticed myself remembering more, either. Rather odd, really. I can remember some of the colour associations I made, but not the context (which is unhelpful), and I can’t remember any of the sound ones. I was expecting this to do /something/, and least :/. I’ll keep trying it – mayhap it’s the newness of the technique.

Try to remember things based on what was said, as opposed to what was written
Paying more attention to what’s actually being said, there’s a large contrast – some lecturers don’t narrate much of anything, some say somewhat more than they write. I still don’t focus for all of a spoken paragraph if nothing is written, which is something I hope to fix, but I least I notice something else is actually being said, now.

An Introduction to Skills Development Part II – Developing Skills?

Follow-up to An Introduction to Skills Development (and the Warwick Skills Portfolio) from Midgley, Christopher - Pointless twaddle and meaningless diatribes

Slightly more than a week has passed, evaluation time! Tutor was Samena Rashid.

P12, slightly modified: When I hear about a new idea or technique, I immediately start working out how to apply it to other situations/problems – the benefits of said idea/technique.
I can now see how some techniques in other fields might relate to a problem I’m working on, but I still don’t wonder what else they could be used for. Most of the time I don’t even recognise other uses within the module itself.

T23: I prefer to evaluate the soundness of my ideas before sharing them.
I..think I’ve said less foolish things recently? This is somewhat related to A13, but I still tend to vocalise my route even as I’m exploring it. I’ll try to ponder a while longer on ideas for this one.

A13: I like the challenge of trying out different ways of doing things.
I’m reading more, if slowly. I can spot different paths to the answer on the subjects where it’s easier to do that (Combinatorics, Stats) but actively avoiding other solutions in some modules, like Analysis, where I stick to my guns no matter how far afield I wind up. This part is also related to P12 – the routes I like to solve things by are rather well defined.

October 20, 2011

Effective Learning Skills? VARK!

Portfolio was on the 19th; tutor is Ceri Marriott.

Completed a VARK questionnaire; can’t say I think the results reflect terribly well (5/5/9/5) – while I am good at learning from reading (and recitation!), I learn better from worked examples and the Kinesthesia score is far too low for that! I’d agree on the low aural score if not the visual – after listening to someone talk for too long I feel my attention waning.

So, points:
1. Take advantage of synaesthesia of memory by associating words with colours/sounds/whatever – give yourself more mental hooks to connect ideas with
Don’t think this one needs explanation – it’s a simple trick. Hope it works :)
2. Try to remember things based on what was said, as opposed to what was written
This is trickier – I think it’s probably a good idea to develop my aural learning style as the majority of ideas are delivered in lectures. I think I’d currently learn the same amount if the noise in the lectures themselves was replaced with a dissonant buzzing so long as the same ideas were expressed in writing. Once again, taking things in in multiple ways should aid memory.

Going to stop there – shouldn’t ask too much at once, I think.

October 16, 2011

An Introduction to Skills Development (and the Warwick Skills Portfolio)

Attended the first workshop on Saturday (15/10/2011) with Samena Rashid as tutor.

Began with the process of reflection, followed by reflection on reflection, followed by brief explanations of VARK and Kolb’s learning styles model leading on to a questionnaire relating to Honey and Mumford’s model. While VARK (Video, Audio, Read/Write, Kinesthetic) aims to match students up with a preferred learning style (innate) so that they can personalise their learning to provide the most immediate benefit, Honey takes [Activist/Reflector/Theorist/Pragmatist] to be acquired preferences and encourages you to improve underutilised styles to learn better through general experience, as opposed to personalising it. From what I saw we take these theories as gospel, and I’m finding it difficult to find literature on the theory itself as opposed to “find your learning style” tests and commercials.

The questionnaire we were given is aimed at business/managerial-type learners, and it shows – some of the suggestions are irrelevant, or at least rather odd, for pure students. As an example, considering only practical items excludes much of pure mathematics (although chasing an example of how pure mathematics can be realistically used, I’ve currently failed to find anything where maths was developed for its own sake and later became practical). Somewhat disillusioning.

Reading through the list of statements and potential strategies to implement so that you find yourself more likely to agree with the statement in order to be a more balanced learner, I find myself questioning – are all these statements positive things you’d want to agree with? (This is aided by the fact that some of them contradict each other!) Additionally, some make assumptions that may not apply (essentially, that the opposite of the statement is true) – “I find rules and procedures take the fun out of things” suggests you try to bend the rules as far as you can, something that I much thought was the whole point.

Still, it looks on the face of it that some of these items are actually self-improvement, although naturally I find myself drawn more to those which most agree with the person I already am, which is rather against the point of the whole thing. There is the added difficulty that for some I feel I already implement the suggestion but still disagree with the statement; while for others I agree with the statement but would find it difficult to implement the suggestion! Still, as self-improvement is the game, let us begin:

P12, slightly modified: When I hear about a new idea or technqiue, I immediately start working out how to apply it to other situations/problems – the benefits of said idea/technique.
Pragmatist is rather a tricky quadrant to go for but I hope I’ve got the gist of the intent here. Essentially, I need to expand my technique library by observing the ‘how’ of the techniques applied in logical arguments / algorithms / relations and checking similar or dissimilar circumstances. As this requires a new idea or technique, I can hardly be specific here.

T23: I prefer to evaluate the soundness of my ideas before sharing them.
Currently I’m somewhat of an Activist-Reflector on this front – as a site I found says, I say something, think about it a bit more, then realise and regret it. It shouldn’t be too difficult to mentally follow the conversation a while longer, checking for flaws – it’s not like I’m losing anything by contemplation.

A13: I like the challenge of trying out different ways of doing things.
Vary my routine somewhat. While lectures are effectively non-debatable (I still think they’re useful), I can change around other free times items. I have books I can read instead of using notes on my laptop. I could also attempt to solve problems in multiple ways, even after I’ve found one working solution.

I’m not sure about doing anything from Reflector – while I think the few remaining points are all positive items, it would slow me down even more than I already am. Also, I’m already at three, and don’t want to attempt too much.

October 09, 2011

Mathematical Statistics A – Example Sheet 1

Thought I may as well put answers to the A & C sections here, for simple-ish reflection. And to notice when I can’t recall how to do things, at all.

QA1: We compute the cumulative distribution function by integrating the density function:
  \begin{align*} F_x &= \int_{-\infty}^x f_x(s)ds\\ &=\int_0^x e^{-s}ds\\ &=\left. -e^{-s}\right|_0^x\\ &= 1 - e^{-x} \end{align*}
We argue Y is discrete because…it takes a finite number of possible values? It’s a step function? I’m not sure. Its support is {0,2} (I hope), pmf and cdf are:
  f_Y(y) =\left{ {1-\frac{1}{e}} \quad (y=0) \\\frac{1}{e} \quad\quad (y=2)\\0 \quad\quad (y \notin \{0, 2\}) \right.
  F_Y(y) =\left{ 0 \quad\quad (y<0) \\\1-\frac{1}{e} \quad (0\leq y<2)\\1 \quad\quad (y \geq 2) \right.
QA2 is simple infinite sum fiddling combined with the identity for e^k.

  Y = g(X)\\ F_Y(y) = (g(X) \leq y)\\ \begin{align*} F_Y(y) &= (X \leq g^{-1}(y))\\ &=F_X(g^{-1}(y)) \end{align*} \\ \begin{align*} f_y(y)&=f_x(g^{-1}(y))\times(g^{-1}(y))'\\ &= \frac{f_x(g^{-1}(y))}{g'(g^{-1}(y))} \end{align*}
By the inverse function theorem (differentiation). Y is continuous as the functions that make up Y are continuous – additionally, as g is an increasing bijection it is continuous and differentiable, and its derivative is always positive, so the pdf of Y is not undefined (g’ always nonzero).

For the last part, simply plug in g(X) = aX + b to what you’ve already worked out, and find Y is the Gaussian (given X is the standard Gaussian).

Find the mgf of the exponential in the normal way; use the same trick as Q4 to find the mgf of S_n. Find the mgf of S_n using the pdf; as they are equal use uniqueness to conclude the distribution is correct.

Integrate F_{S_n} from t to infinity; receive sum from 0 to n. Observe that as the number of photons detected must be an integer, P(N=k)=P(N<k+1)-P(N<k). Plug in your previously calculated values for the answer.

October 2011

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  • Nice proof! Does this mean you're going to specialize in analysis and differential equations next ye… by Nick on this entry
  • Hi Chris, It was most interesting to read your various reflections – thank you for sharing them. I'm… by Ceri Marriott on this entry
  • Feel free. Chris by Christopher Midgley on this entry
  • Hi Chris This is an honest final entry for the WSPA. Im glad that you have found the WSPA journey wo… by Samena Rashid on this entry
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