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The Next Level- 2

February 23, 2010

Previous blog post: Covering Basics
Other posts in Quantitative Finance

Continuity is the essence of time.
So while we like continuity in every form of life, why do we hold back in case of random variables?
Lets us go a step ahead and try to work our head around continuous random variables.
Its simple to start with.
Say, you have got a top(not tank top, you maniac, a spinning top). And you are going to spin it, and drop it on a circle of radius r. Now, it doesnt end there. You want to check, what is the probability with which, it will land on the top half.

In case you forgot, how a circle looks like

So here, the net thing doesnt change at all.

How you approach the problem is the same old thing: Determine the sample set.

The sample set is the entire circle. You cant count the outcomes here, but still you can “estimate” it . Its simply the area under the circle.

Sample Set: \pi r^2
Favoured Outcomes: \frac{1} {2} \pi r^2
Hence the probability is : 1/2;
Pretty simple, but note that we didnt really count anything. It can also be discerned and arrived at this number by checking out, that there are two ways, going by the way, we have the outcomes. The top lands on the top half or the bottom half. 2 Outcomes. 1 favoured.

The next point is, you can actually estimate anything in real life, by actually “estimating” the area under the curve.

Let us leave this thread for awhile here. Let us comment a bit on an observation.  There is an observation, that say, you have a well sampled population (though it has connotations, but for all practical and cognitive purposes let us, just read it as population) and you try to measure/ observe some measurable attribute, like… height?, or for that matter, marks in a test, or speeds while running on a track?

People(who had no other jobs, other than keep measuring this stuff) saw, that there was some strangeness about the entire observation.It looked like, majority were average, a few below average by a certain degree, still fewer even below the previous group and so on and so forth. And this, was repeated on the other side, i.e a few people were just above average, still fewer just above this group, and so on.

Coarse Granularity

In essence, its a staircase type distribution, when you plot on x-axis the measure of the attribute falling in a particular range and on y-axis the number of students/people satisfying that condition . The first half is a rising staircase, the second half a falling one.

Now the sexy thing about it, is, if you measure even more accurately or divide the ranges even more finely, you can see that, the entire staircase morphs into something called bell type distribution.

Fine Granularity

Mathematicians, in fact, Gauss, said, that this is a universal distribution found almost everywhere and called it Normal Distribution.

This Normal Distribution has an average, also called the first moment.It has a standard deviation, that is, the average of (sign irrespective) deviations from mean, also called the second moment. So intuitively, it will talk about how thick the stem is, or how lean the stem is. Higher the deviation, higher thickness.

And yes, it isnt any surprise that the area under such a distribution curve will yield the entire population. Now to keep it generic, we have scaled the entire curve such that, the area under the curve is 1. Also called the zeroth moment.

Oh yes, btw, did I tell you, that within the first one standard deviation on both sides of the mean, cover 68% of the entire population. That is, majority of the population is just marginally better or worse off. And 99% of the population falls within two standard deviation away from the mean. Come to think of it. 99% of all people you meet, are just a bit better than you, or a bit worse than you.

It isnt much fun, until you try to take something away from it for your own immediate development.

Two things.

a. Majority of the people around you, the people you meet day in and day out, sit, talk and eat with are just that. Average. You will have average success, average wealth, average peace, average happiness and everything else average. Want to move to the rariefied world of spectacular success? Stop being average? How? Start shunning every bit of ol’ habits you picked up, while you were busy running with the crowd. Remember, nobody beats the crowd;the crowd beats itself to pulp just by being that,  A MINDLESS CROWD. So break out of the mould.

b. Read this, the Psychology of Those Who Win, by Dr. Brett Steenbarger. Catch up on some reading on Gaussian Curve. We will try to work up more on that part very soon.

P.S: No equations in this post, for those who really are dying to see some equations, read up the wiki enty of Gaussian Distributions. In the next blog ,we will be working with some equations, Bernoulli Distributions etc.

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