Underlying Pattern Governing The Prime Divisors

1 mainWhen we think of prime numbers, the first thing that we tend to associate them with is randomness. Prime numbers are scattered all over the number line and there is no fixed formula that can tell you when the next one is going to occur. This has been used heavily by mathematicians and cryptographers to develop security systems for internet, banking, communication, and so on. Coming to the topic at hand, we are going to talk about prime divisors of a given number. Prime divisors of a number are divisors of that number that happen to be prime numbers. Big surprise, right? Alright, what’s so interesting about them?   Continue reading

Gaussian Mixture Models

mainLet’s say you have a lot of data and you want to estimate the underlying statistical model. Wait a minute, why on earth would I care about that? Well, if you estimate the model, then you can analyze unknown data that is not under our control. Some of the common examples would be weather estimation, facial expressions analysis, speech recognition, share prices, etc. Coming back to the estimation problem, the simplest thing to do would be compute the mean and variance of this data, hence getting the governing distribution. But what if there are multiple subgroups in this data? As in, how do we detect the presence of subpopulations within an overall population? Even though the data points belong to the overall data, we need to understand the different modes inside the data. How do we go about doing this?   Continue reading

Interpretation of Gaussian Distribution

When we deal with large amount of data, we can’t have specific rules for each and every instance. We have to come up with a model which defines the whole data. This model can then be used to analyze unknown inputs. More often than not, the data has some underlying pattern. When we think of a model, we extract specific characteristics from the data and come up with a formulation which best explains the behavior of the data. One of the most frequently occurring pattern is the Gaussian Distribution. It is used almost everywhere in science and technology. But what is it exactly? Why do we need it?   Continue reading

Probabilistic Randomness Of Stochasticity

Do you see what I did with the title there? Anyway, you must have heard the term ‘probability’ being used around you. People use it in different contexts and in different forms – “What is the probability that Spain will win the next world cup?” or “I will probably finish reading the book by midnight” or “It’s quite probable that she won’t return until tomorrow”. When people talk about probability as a mathematical concept, all they think of is the percentage chance of something happening. But is that all there is to it? If that is the case, then why did they have to dedicate an entire branch of study to this? Probability theory is much more than just calculating the likeliness of something happening. It’s used almost everywhere, by almost everyone, for almost everything. Surprised? Well let’s find out then. Continue reading