Unruled Notebook

Entries from May 2008

Probability Density Function

May 28, 2008 · 3 Comments

Discrete Random Variable, their expectation, variance and standard deviation were explained earlier. In general, three kinds of probability distributions – binomial, normal and Poisson (also known as rare event distribution) – govern events that are described by such random variables.

Continuous random variables are different. They can take an infinite number of values within a given interval. We cannot list out all these values simply because we cannot mark two neighboring values for a continuous variable. The curious thing is the probability of a continuous random variable taking a concrete single value could be zero. Of course, we need to overcome this situation. Let us see how.

We are now dealing with a continuous random variable. So, we shall skip the blog popularity example that we used while discussing Random Variables. The analogy gets tedious. A roulette wheel is more manageable. Suppose our continuous random variable x is allowed to take any (real) value including all the decimals in between the marked integers in a roulette wheel. So after a spin and stop, where will the roulette wheel pointer stop? Somewhere for sure. But can we predict it? That is, what is the probability that the pointer will stop at a certain (real) value? In other words, after a roulette wheel spin and stop trial, what is the probability that x will take a definite value? The answer is zero. Yep.

The roulette wheel is essentially a circle, unless it is rigged by one of those guys from the Ocean’s Eleven series. So for convenience let us divide the circle (in which the real values are marked) into fifteen equal tangible sectors. Now we know after a spin-stop trial the roulette pointer will stop in one of these sectors – assuming the dividing line is an imaginary one of zero width. Obviously, the sectors are of \Delta x = 2\pi R / 15 length each. So we can ask the question what is the probability that the roulette pointer will fall within one of these fifteen sectors (say, between x and \Delta x ) and the answer would be \Delta x / 2 \pi R = 1 / 15 . That is, the probability that a continuous random variable ‘x’ would fall within a range x and x + \Delta x in this roulette wheel example would be

p_x = \Delta x / 2 \pi R

If we now reduce the \Delta x , i.e., if we increase the number of sectors the roulette wheel is divided into, obviously, the probability that the pointer will fall within one of the sector will correspondingly reduce.

Extending this argument to a situation with \Delta x \rightarrow 0 , we would have the pointer located at x (no more a ‘range’ as \Delta x \rightarrow 0 ). Here, the probability as defined in the earlier equation, is zero. So as said earlier, the probability that a continuous random variable x would take a definite value is zero.

But we did end up with a definite value, which means this is not an impossible event (for which is what the probability is zero). So what did we do wrong? Just the limits.

This is where the concept of a probability density function comes in.

Assume a point P located somewhere inside a solid block, labeled with a position vector r. Let a volume V envelope the point fully and let this volume contain mass M. We shall now reduce the volume V in steps as V_1, V_2, ... such that the point P is still enclosed within each of these volumes. Correspondingly the mass of these volumes would also reduce to say M_1, M_2 and so on. When the volume is reduced to zero, we would have reached point P, but with a corresponding mass M = 0. In other words, the block (or body) of a finite mass seem to be made up of an infinite number of points of zero masses.

To strike a parallel, the non-zero probability of the roulette wheel pointer would stop at any one of the roulette wheel value is a sum of the infinite number of zero probabilities that the pointer will actually stop at an exact value within the considered range.

This analogy gives rise to the probability density function. Instead of dealing with the zero mass of body points, we define the concept of a density that is non-zero even at a point.

If \Delta M is the mass of the volume \Delta V within which we have the point P located at a position vector r, then the density at this point is given by the ration of the \Delta M and \Delta V as the limit of \Delta V tends to zero, written as

\rho (r) = lim _{\Delta V \rightarrow 0} \Delta M / \Delta V

If the volume \Delta V is small enough, we could write \Delta M \approx \rho (r) \Delta V . The mass M of a body occupying a volume V then would be

M = \int _{V} \rho (r) dV

where the integral is performed over the volume dV.

We can borrow this concept back to our roulette wheel, continuous random variable and probability. So when dealing with continuous random variables, probability theory uses the probability density function rather the probability itself.

If f(x) is the probability density function of a continuous random variable x then it can be defined as

f(x) = lim _{\Delta x \rightarrow 0} \Delta p_{x} / \Delta x

Here \Delta p_{x} is the probability that the random variable will take a value between x and x + \Delta x . So if we turn around and ask for the probability that a random variable will be between say x_1 \) and \(x_2 , it is given as

p = \int _{x_{1}} ^ {x_{2}} f(x)dx

Observe here if above integration is performed over the entire range of values the continuous random variable can take, it would be equal to one (the probability of a sure event). In our roulette example if we ask for the probability that the pointer will end up after a spin-stop trial in at least one of the sectors, then the above integral needs to be evaluated between 0 and 2\pi R and the result would obviously be equal to one. In general, if we assume a large range for the random variable we could write

p = \int _{- \infty} ^ {+ \infty} f(x)dx = 1

Specific to our roulette example, we could observe that the probability that the pointer will stop within one sector is independent of what sector it is or the value of x itself. This simplifies the probability density function integral as

\int _{0} ^ {2\pi R} f(x)dx = f \int _{0} ^ {2\pi R}dx = 2\pi R f = 1 \text{ (OR) } f = 1 / 2\pi R .

This is similar to the block case, when it has a uniform density so that it becomes independent of the position it is measured (\rho = M/v ). In general, the density \rho (r) varies from point to point in a solid and so can a probability density function of a continuous random variable.

For completion, the expectation and variance for a continuous random variable are given by

E(x) = \int _{- \infty} ^ {+ \infty} x f(x)dx

and

var = \int _{- \infty} ^ {+ \infty} (x - E(x) ) ^ 2 f(x)dx .

Compare these expressions with the corresponding ones for a discrete random variable.

[Surprisingly, I couldn't find in the internet a write up on why we require a probability "density". The many sites that a google search takes me doesn't explain this. Of course they give the correct formula and even tutorials. Hence this write up. Let me know if you locate any useful links.]

Categories: Maths
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Geek Rainbow Date

May 25, 2008 · 3 Comments

Possibly some of my students will immediately latch on to this one I made. For those who need some clues, see below the cartoon.

The hex chart clue is here; The geek tag was an idea improvised from Rubbersquid blog; The black and white picture is of a Rainbow at Portobello from jimbodownie’s flickr storage modified to suit the need; the girl is from the wrapper of the Adventures of the White Girl in her Search For God (1933) by Charles Herbert Maxwell, taken from here. The rest is deliberately left unexplained.

Enjoy the summer vacation geeks.

Categories: Cartoons
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Blogging is Academic Time Waste

May 23, 2008 · 5 Comments

Excuse me Sir…

Yes.

You are this BIG Prof.?

May be…

OK, so you aren’t.

Er, I didn’t say so…but why are you asking?

Do you blog?

NO.

Why?

Why should I?

Er, isn’t it fun? Isn’t it useful? Do they not help you in your research? Don’t you think discussion is important while doing Science or practicing any subject? Isn’t internet a democratic medium that provides this opportunity for you all for free?

NO.

OK, do you read blogs?

Waste of time.

Not even the ones that are related to your field of research or reading interest?

Most of them are crap.

So you are aware of blogs after all?

Well, yes…

Aha! For a moment I thought you are a BIG Prof…

@#$%^%$##@

Categories: Micro Muse
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On the Effect of Chennai Summer on Shaving Cream Containers

May 21, 2008 · 5 Comments

Abstract: [This is a technical note. It doesn't require an abstract]

Introduction:

Shaving is an act of curtailing growth since time immemorial. For instance, it is reported in Valmiki Ramayana [1] that even Rama shaved, upon his return from vanavasa (exile). Shaving creams are recent reluctant aides to literally cover the spoils of this ancient act. However, because of their high “free energy” (as in they behave the way they like when set free), they are usually contained using metal containers sealed with thumb depressors and marked “shake well after use”. This study investigates using experiments, the effect of the Chennai Summer on such containers. The outcome should somehow affect directly shaving in general and shavers in particular.

Experiment:

Experimenters were conducted (performed, as suggested by the reviewer [2]) in uncontrolled conditions in a typical Chennai home well soaked in Summer. The apparatus consisted of Shaving Cream container, Flat table and a High end, you-Poor-Indian-Guys (pig for short)-can’t afford, NRI gifted, US Junk Mall discount sale purchased, real time reality digitizer cum pixelizer.

The experiment is performed with the rest of the apparatus kept vertical but the digitizer kept horizontal , a direct result of inefficient lab technicians who don’t realize the big picture of where their experiments fall into in the vast cauldron of human refuse.

Results:

[Experiment Results]

Discussion:

Chennai Summer is very hot, unlike Chennai Winter, which is just hot. This is only expected given the seasons of Chennai – Hot, Hotter and Hottest.

[Image Courtesy: Globalwarmingart.com]

Given such favorable conditions, animate or inanimate, things behave bizarre. This is clearly shown in the above experimental results.

It can be observed that in Chennai summer, shaving cream containers froth and die a painful death.

The experiment is unrepeatable as things behave unpredictably in Chennai summer. Further experiments (with coke and pepsi containers shaken well before use) are necessary to fritter summer time this way, as thinking at super heated neuron conditions becomes thoughtless.

Conclusion:

Its damn hot in Chennai, you know. It is May and it is already 108 Fahrenheit (or is it Celsius).

Acknowledgment:

I thank the prudent reviewer who pointed out [2] that there is no apparent connection between the title, content and conclusion of this note and once it is established it can be considered for rubbish. I agree the reviewer is correct in the assessment but misses the point of the note completely. I take extreme pleasure (paroxysms of joy, to use Ramesh Mahadevan’s [3] patented phraseology) in not trying to establish any connection whatsoever. Once done, as the reviewer points out, this note would become rubbish. I have answered the reviewer in kind [4].

References

[1] Ramayana, by Valmiki, Parchment Publishers, circa 20000 BC give or take a few centuries, [Original out of print - ask Advaniji for whereabouts]

[2] Reviewer (name kept anonymous) report for “On the Effect of Chennai Summer on Shaving Cream Containers” [declined for acceptance - see [4] below]

[3] Ramesh Mahadevan, father (and mother) of all Indian internet humour (notice the u).

[4] the author reply in full.

Categories: Muse
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Your Review is Hereby Summarily Rejected

May 21, 2008 · 8 Comments

From

The Editor

The Highest Impact Factor Journal in Your Field

To

The Anonymous Reviewer

School of Irrelevance

(rest of the address is kept anonymous)

Dear Anonymous Reviewer,

The author reports have arrived for your review comments submitted an year back.

Based on these author reports I have come to a decision about your review report on the Author’s Dear Paper of Two Year Labor.

It is with extreme poignancy I write to inform you that the authors, after sitting on your review for an year, have summarily rejected your review report for want of better content worthy of publishing consideration with us.

In this context you should note that, your submitted review report claimed the non-existence of the author’s work based on purely personal grounds argued with enough weight in sentences like “I don’t like the figure”, “the manuscript is in bad shape” and “I don’t like the author’s name as I am unable to remember the spelling while typing”.

Also, given the busy schedule and extreme duress the authors are subjected to and given the nature of the thankless job “paper writing” is (as against peer reviewing), I am sure you would be thankful for such a quick (only an year) turn around time for the author reports on your review.

You may choose to resubmit after writing a totally new reviewer report, which will go through a full authorship process. I await your resubmission.

In any case, thanks for choosing The Highest Impact Factor Journal to submit your reviewer reports.

Signed

Editor

Enc.

Author Reports [unsigned copy in triplicate]

———-

[Disclaimer: Just a fantastic thought what if we reverse the entire peer review process. Be kind on me. I have been and continue to play the author and the reviewer and both tasks are tough. But I am sure you will agree peer review process is still the better method for publishing qualified research [1]. I abide by it.]

Reference:

[1] follow that link to read the entire latest report on peer reviewing by the Publishing Research Consortium.

Categories: Academics · Micro Muse
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