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.

Exponential Distribution

The gnuplot script used for creating the image is as below

set terminal png
set output “exponential distribution.png”
ymin=0.0
ymax=3.0
set yrange[ymin : ymax]
xmin=0.0
xmax=4.0
set xrange[xmin : xmax]
set xlabel “x –>”
set ylabel “probability density –>”
set title “Exponential Distribution”
expopdf(b,x)=(b)*exp(-x*b)
plot expopdf(0.5,x)  title “alpha = 0.5”,\
expopdf(1,x)  title “alpha = 1”,\
expopdf(2,x)  title “alpha = 2”,\
expopdf(3,x)  title “alpha =3”
It is easy. Isn’t it?

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

German Tank Problem

There is so much to know in this world!!!

How will you calculate the number of tanks that your enemy  is producing? You have a set of serial number of the tanks which are numbered from 1 to n with m being the max and k samples? well you use the formula

Max tanks = m + missing numbers/k.

A derivative of this method was used by the allies during WW2 to estimate the German Panther production (and later during the Cold War to estimate Russian Tank productions). The method was not as simple as given above and is detailed as usual in the wiki article.

http://en.wikipedia.org/wiki/German_tank_problem

I have always believed that you can get amazing information from trivial data by proper analysis. Even absence of information can provide a huge amount of information!

You have a library card of a book on New Mexico from a University library containing the list of people who were issued the book. If you are a German spy what information do you get from it? Well you can come to know the location of the top secret Manhattan Project.

You would notice that the list contains names of physicists and engineers and you will also notice that most of these people have disappeared from the campus. Now you know that these “disappearing” people have most likely gone to New Mexico and you will concentrate your attention there to find out about the Manhattan Project.

http://en.wikipedia.org/wiki/Stanislaw_Ulam#Manhattan_Project

😀 You come to know new things every second!

http://en.wikipedia.org/wiki/Stanislaw_Ulam

Ulam was among the first to refer to the technological singularity—and possibly the originator of the metaphor itself—in May 1958, while referring to a conversation with John von Neumann:

One conversation centered on the ever accelerating progress of technology and changes in the mode of human life, which gives the appearance of approaching some essential singularity in the history of the race beyond which human affairs, as we know them, could not continue.

You are watching a test of a surface nuclear test. You tear up a piece of paper and dribble the confetti in the air as the shock waves hit. What are you doing?

You are trying to determine the yield of the nuclear explosion!

http://en.wikipedia.org/wiki/Enrico_Fermi

Fermi was present as an observer of the Trinity test on July 16, 1945. Engineer Jack Aeby saw Fermi at work:

As the shock wave hit Base Camp, Aeby saw Enrico Fermi with a handful of torn paper. “He was dribbling it in the air. When the shock wave came it moved the confetti. He thought for a moment.”Fermi had just estimated the yield of the first nuclear explosion. It was in the ball park.[10]

Fermi’s strips-of-paper estimate was ten kilotons of TNT; the actual yield was about 19 kilotons[11][12]

http://www.lanl.gov/history/atomicbomb/pdf/Enrico%20Fermis%20Observations%20at%20Trinity,%20July%2016,%201945.pdf

Pi

How do you calculate the value of Pi??

You draw a circle inscribed in a square. The ratio of the area of circle to that of the square if pi/4.

Now take some grains of rice and randomly (Randomly is the most important part) throw onto the figure that you drew. Since the throw was random, the number of grains inside the circle to the total number of grains inside the figure would follow the ratio pi/4. Perform this experiment sufficient number of times (Asymptote :D) and eventually you will get the perfect value of pi. Aint it simple??:) This is an example of Monte Carlo method which talks about using random numbers to simulate complex situations. More about it later.

Now the problems in this method:

1. You have to perform this experiment a huge number of times to get the perfect value of pi and get the average value. Ideally an infinite number of times. (This is my understanding).

2. Creating a perfect random throw is impossible. We havent been able to create a perfect random number generators yet. All those random functions that we use in our “unbreakable” ciphers are just pseudo random numbers. Maybe the incidence of cosmic radiation on any square meter of earth per second would come as close to uniform random number as possible.

Until this “practical” method is perfected, we will have to deal with the mathematical methods to find the value of this amazing number.

Wiki has a very good GIF image of the meaning of pi which should be taught to each and every child before he is told to mug up the formula for circumference and area of circle. http://en.wikipedia.org/wiki/File:Pi-unrolled-720.gif

Most images in Wiki have CC sharealike version 3.o license. I dont know what needs to be done to legally use those images. 🙁

Feynman Point: Richard Feynman, one of my favorite physicist, has given his name to a wonderful anomaly in the digits of pi.

http://en.wikipedia.org/wiki/Feynman_point