Tuesday, November 23, 2010

Mathematics of an Exam

Guess what. I am nearly done with another set of exams. No matter how much you try to shoulder arms at them, they make you go on the back foot and defend. Now that Im done with the theory exams, I am freely able to punch my practicals through the covers! No worries. My timing in the practical viva is as immaculate and as elegant as those of Sachin Tendulkar on the field. (Forgive me for a cricket like opening, but im watching India thrash NZ by Innings and some runs, so the effect rubs off..)

Now then, coming back to the point, as this was my 7th semester in the college, I have given a total of 14 exams plus a number of 10 mark tests. A large enough sample size has been generated through the experience of me and my friends to draw out the following conclusions about the time near to exam. These will be known as the Datar-Devi Approximations, after its researchers, Datar and Devi.


To start of with. Let 'T' be the time left for an exam. This is usually about 2-3 days for a typical COEP paper, for which one can actually study. For simplicity we shall assume that T is a rectangular distribution. from -T to 0.
However, we never start studying at the start of the rectangular function. Studying is a statistically boring process. We usually start our studies at an arbitrary time instant H, after -T. Again, this will be a rectangular distribution over T.

Now, as it is statistically observed, attention span is dependent on H. let Asp be the attention span.


This will give you the attention span that you will get, depending on when you start studying. This is a critical factor for getting good marks. ideally, toppers have H close to T.

The second important point that is under consideration is Number of calls you make asking your friends "How much timepass you have done?" this can be denoted by N(h). Number of calls before exam are way more that those on ordinary days.
N(h)=e^2kh.. k= Phony constant. the derivations for above terms are extremely complex and is beyond the scope of timepass. So, it has been left to the Chinese to come up with the proofs for the same.

The third important factor is time you end up swearing at and cursing your Professors. Now, as the trend is, we curse our profs as being useless and incompetent much more during exam period than normal.This analysis is extremely interesting, and thus, has been discussed below. Now, not all professors are bad. some are actually superb. so lets have a sample size of N professors.
Let us consider r activities- such as teaching, setting papers, fluency in speech etc.
let 'p' be the probability that the teacher is good.
so, (1-p) is the probability that the teacher is rubbish.
then, assuming a binomial model and taking average over given number considered, we can say that the swearing coefficient is






The fourth point considered in this treatise is based on pure statistics. The point is know as Chinmays attraction law. "As exam time nears, in D Days, your feelings towards an arbitrary girl goes on increasing to the exponent".Here, γ(gamma) is the attraction probability. γ is dependent on several factors such as the nature of the person, the college where he is studying etc. If γ is the attraction probability, then, the attraction coefficient can be given by  




However, in colleges like COEP, as there are very few good looking girls, and the fact that the college is filled with nerds, γ is negligible. So taking a Taylor series expansion, and ignoring higher order terms of γ ,
we can say,





The next important point is concentration span. You tend to concentrate very little while studying. breaks are frequent. Let B be the number of drinks/snacks break you take in between a span of 3 hours- a statistically important variable as all papers are 3 hours long. Then, concentration span, CSmax will be:
CSmax= 3hours/B.
Usually this value is quite low, owing to the statistically high value of B (12-15 in normal cases).
The analysis of B is also again very complex, and has been left to the Chinese.
One thing we know for certain is that CSmax is inversely proportional to t, the time before exam.

The next factor is the ratio of useful study to time spent on Facebook. This is called as the phace inverse ratio.

Now, PIR= (T.i.m.e.o.n.a.b.o.o.k/T.i.m.e.o.n.f.a.c.e.b.o.o.k.).
cancel the common terms in the equation, we get,
PIR=(1/face)

It has been statistically proven by the likes of our peers Shriram Kardile and Sahil Patwardhan, that time spent on Facebook is roughly 5 times the time spent on reading a book.
so, it can be concluded that PIR=(1/5)
(from their publications Facebook is more fun that studies. Kardile,Patwardhan and Datar.)

Ok, so I  guess these many factors are enough to formulate the time remaining for the exam T.
We can conclude, that T, the time to exam is directly proportional to:
1)Attention span
2)Swearing time
3)number of calls
4)coefficient of attraction
and is inversely proportional to
1) Concentration span.

Thus, we can mathematically conclude:




Introducing the constant of proportionality, the PIR to the equation to get rid of the proportionality sign and substituting the values for respective coefficients, we get


  Thus, finally we have arrived at the examination formula.

Original research carried out by:
Chinmay A Datar
Anant M Devi.
Helping Hands:
Sumedh Dhabu
Virendra Kakade
Abhishek Ambede.

References:
Probability and Random Processes
-Stark and Wood. (3rd years, see, its not that bad a book. has many useful applications afterall)
Why facebook is better than book
-Sahil Patwardhan, Ajinkya Rao
Facebook to book ratios
-Shriram Kardile.

We thank all our friends for supporting us throughout the research, and are hopeful that our efforts will be awarded at the IEEE conference of vadheevpana.
-P.S- mathematical errors have been deliberately introduced to hide original equations and research.
-Authors.

6 comments:

  1. Okay so I did not have to read the entire thing to say that ok,chinmay datar is back to writing what he writes best - ultimately vadheeev posts!
    changlay changlay..i'd rather read these than your dedicated to school friends or music posts..

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  2. khara ahe are. ya end-sem la tar kahar zhalay timepass cha! :P ;)

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  3. Nice! You could give S.B. Patankar a run for his money :D

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  4. chinya 1 no.!! phony constant lai avadla!!

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  5. hahaha just too good!
    Introducing the Chinese thing was brilliant! :D
    i might as well make a concurrent model with boundary conditions as that in VIT...

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  6. legendary!
    awesome research!the potshots at the chinese were great,though i did'nt get why the chinese in particular...there's still scope for further research,find the limiting values for T, it might be useful for future generations of suckers who commit the blunder of joining COEP :)

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