MATH MAANIAC

             The word  " Probability " often comes into our day-to-day life referring to, " What is the probability of some random event ? " simply saying what at most chance the event to occur.            In the last two blogs " The Axioms - Probability " and " What is the probability you will open this blog ? " we see how probability arises over time.         But the question is if the event space contains more than some finite or infinite elements, How will we calculate the probability?       Here the role of Probability Distribution came into the story. So instead of calculating, we evaluate how the probability is distributed over the region.       First, take the simplest example, " Tossing a coin ", so by the axioms of probability we have here two event spaces called head and tail. The probability of getting the head is 1/2 and the same for the tail.  ...

                The word  " Probability " often comes into our day-to-day life referring to, " What is the probability of some random event ? " simply saying what at most chance the event to occur.                 In my last blog " What is the probability you will open this blog ? " I told you, why for a long time,  I have been in a dilemma, about why probability is so useful , my counter-argument was, "Probability tells us about the most probable event to happen based on the available ones, it does not give us the guarantee that the event to be happened for sure, then why it's so overrated ?" and I give you a classical definition of probability.             Although the classical definition easily gives us the probability, it has some ambiguity . 😢😢            1)  It is applicable when the total number of events are finite. ...

                        The word  " Probability " often comes into our day-to-day life referring to, " What is the probability of some random event ? " simply saying what at most chance the event to occur.             For a long time, I have been in a dilemma about why probability is so useful , my counter-argument was, "Probability tells us about the most probable event to happen based on the available ones, it does not give us the guarantee that the event to be happened for sure, then why it's so overrated ?"                      I remember one day I was standing at the junction of three roads, from one way a car was coming, so the question is what can happen next? I assume the driver does not have the same question as mine so she would not stop the car and think, it's a three-junction road what does the boy do next? 😀😀 So sh...

           In my previous blogs " TOPOLOGY - KING OF MATHEMATICS " & " Topological Space " we came across Topology's beauty and discussed its properties. Now we flash on "Topological Surgery Theory" - a technique used to do surgery (cut - operation - join) on manifolds.      To remind you in simple words manifold refers to any object , formally saying" A manifold of dimension n is a Topological space M which is locally homeomorphic to n-dimensional Euclidean space, Hausdroff and has a countable basis. In topology, surgery theory is a procedure to transform one manifold into another manifold in a controlled way.  There are different kinds of surgery on a manifold.  The formal way is to identify an embedded structure X on the manifold M that we want to cut, then the embedding φ : X → M To exclude it, do the cutting operation M \ (int(φ(X))) where int is the interior of a set. For gluing back (if, ∂X ...

        In my previous blog   "TOPOLOGY - KING OF MATHEMATICS"  , talked about geometric shapes and saw how "Distance" is the basic tool for the study of "Metric Space" and also saw how removing the notion of distance lead us to the more general study of spaces called "Topological Spaces".        The goal of today's blog is to understand the Topological Space more formally and challenge our normal understanding to an another level.       As in metric space main key-tool is distance, here it is neighbourhood.        First, we can start with what we mean by neighbourhood, its simply mean surrounding. If you choose a point and a r >0 radius circle around that point then that also a neighbourhood mathematically denoted as B(x,r) where x is point and r is my radius, this neighbourhood called r-neighbourhood. We can see one thing that if we choose r-neighbourhood the study came to study same as m...

         In my last blog on "Measuring a set", we see that how we can generalized the idea of length, area, volume for an set and we define special kind of set notions like sigma-algebra, algebra and define an arbitrary measure on that set we called that "Measure of a set."      As beginning of the last blog I told that our generalization should be same as our existing idea of length, area, volume. The  Lebesgue Measure on  ℝ  is a generalization of length.            So there is two question, 1) Any subset of   ℝ  we take, can we measure it ? 2) The measure of the set and the length of the set is same ?        We know that, if   ℝ  be my set then P( ℝ ) is the set of all subsets of  ℝ . Let's see we can measure any element of P( ℝ ) or not. To do that we first define equivalence definition of Lebesgue Measure on  ℝ .     Lebesgu...