# Category: Mathematics

## Information Age

Todays age is sometimes called Information Age. Somehow I feel that using that terminology for any period of time is incorrect. Every age is and was an Information Age. Information was priceless a thousands years ago as it is now. The only difference that I can see is that we are able to make sense of the information much more correctly and faster than ever before.

All our decision making depends on information. This relation between information and decision making presents some interesting points. Consider a simple game of Tetris. In our version of tetris we have only two types of blocks, a one square block (1s)and a two square block (2s). The pile at the bottom has two slots in which we can fit 1s (left slot) or 2s (right slot). While playing the Tetris we make the decision of where to put the shape that next comes in. Suppose you need to make the decision of where the incoming block goes without even knowing which block is the next one. We can either choose the left slot or the right slot and it will make no difference. Actually it does. Generally when we have to make a decision based on a yes/no information we assume that the probability is like a coin flip. Exactly 50%. Which is not always the case. Thus, it is possible that choosing the left slot all the time gets you a better result than 50%.

This is an example of decision making without information. The information we need to make the perfect decision is to know which is the next block. We make 1 bit of decision using 1 bit of information. Real life is not so simple, we make 1 bit of decision using a huge amount of information (e.g., hold or sell the stock). Now let us assume that we have a little more information about our Tetris game. We know which was the block that came in the previous round. It was a 1s. We also know that the probability of a 2s following a 1s is 0.9. Thus knowing that the previous block was 1s gives us a 90% surety that we can predict which the next one would be. This is not perfect but it is better than nothing.

Now suppose you have another kind of information with you. You have the source code of the algorithm that generates the block. Now you don’t have to worry about any probabilities, you can always know which the next block would be.

Thus we have a classification of information

- Useless information. Strictly speaking this is not information.
- Information that helps us make the decision with 100% certainty
- Information that helps us reduce the uncertainty in our decisions.
- And a Metainformation that helps us generate the information we need.

Thus not all information are created equal.

What do you call it when you make a decision without all the information or the Metainformation? You call it **Risk**. This is the basis of all the risk analysis and risk management principle. You do not have the adequate information and hence you prepare some bulshytt (Note that this word is different from the popular Bulshit) to try to help you make decision.

The reverse of this decision making process is also true. When you know a decision, it is quite possible that you can derive the information that led to the decision. This is true even when you do not take a decision. **Absence of information is also an information.**

Under a rational assumption, if we have 100% information we can make the correct decision and there is no involvement of subjectivity in it. This is rarely the case. This leads to what is called as Bounded Rationality in management parlance. That is we make decision with whatever information we have available to us at the moment.

Lets get to know** Efficient Market hypothesi**s now. EMH or at least the stronger form of it asserts that market prices instantaneously reflects all the available information. There are countless ways in which this seems and is wrong. Technical Analysts believe in EMH but also believe that EMH does not affect the market instantaneously but instead happens in trends. This gives the Technical Analysts the tiny bit of time to get into the trend ahead of the others.

Information has a huge impact on how you become a Goldman Sachs. What if you accept EMH and have access to that self correcting process of the market a tiny moment of time ahead of others. Well, you earn millions out of it. This is called as **Flash Trading**.

In flash trading, exchanges gives you access to the trading information 30 milliseconds ahead of the general public. Thus if you have a sophisticated trading algorithm and access to this information, you can identify the pattern the market is taking to adjust itself to all the information that is available to it. So now you have access to the decisions that were made but may be not the information that resulted in it. You can utilize this “decisions information” and place your decisions into the queue and earn profit. Many markets have stopped allowing flash trading, but you never know what other avenue of information use is still open.

References :

http://www.nytimes.com/2009/07/24/business/24trading.html?_r=1

## gnuplot

gnuplot is a non GPL (uses its own open source license) software program that can be used to plot complex mathematical graphs. gnuplot is a must software for any person even remotely interested in mathematics and statistics. I believe that all school children should be introduced to such amazing softwares as early as possible so that when they are ploughing through those difficult trigonometric formulas (formulas and formulae are both plural forms of formula. The built in Chrome dictionary, however, is not recognizing formulae. :s ), they do come to know about what those functions represent.

Below is the representation of Exponential Distribution that I created using gnuplot.

The gnuplot script used for creating the image is as below

## Bean Machine

Wiki says

The **bean machine**, also known as the **quincunx** or **Galton box**, is a device invented by Sir Francis Galton to demonstrate the central limit theorem and the normal distribution.

## Probability functions

A probability distribution function just tells the probability of occurrence of an event E. P(E)

The mathematical way to describe this probability distribution function(pdf) for continuous variables is the probability density function or simply density function. f(x)

A probability mass function(pmf) describes probability distribution function for discrete variables.

A pmf tells us the value for a discrete point while a pdf for x, tells the value in the infinitesimal range [x, x+dx]. Thepdf for exactly x would always be 0. F(e)

The cumulative probability density function gives the probability for value at x or less than x.

Thus

P(E<=t) = F(t)=Integral from – infinity to t of f(e)de

PS: Calculus is also added to the list of things that I have to learn