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Why Is There An Extra Factor Of 4 Times Pi On The Right Hand Side Of Gauss

Why is the normal distribution important?

Why is the normal distribution consider as the most important probability distribution?Suppose you have a large sample from a non-normal distribution. Now suppose you estimate a parameter of the distribution such as the mean or the median. This statistic itself has a distribution. That is, if we were to keep repeating the sampling and estimation process we would get a set of different estimates and these have a distribution. (We don’t repeat the sampling process, we can deduce the distribution from theory.)If the sample is large enough this sampling distribution will often be approximately normal.The conditions for this to occur are that the estimate is some sort of sum, or approximately a sum, or a measure near the centre of the original distribution.This could require a sample much larger than we are willing to use, so we usually require that the original data have a distribution that doesn’t produce many outliers (very different values from the rest) and is not too skewed.

Mathematics: What is the most beautiful theorem proof, and why?

Though their are lot of elegant proofs in Mathematics, here I am posting two of those.1. Proof by contradiction of an infinite number of primes, which is fairly simple:Assume that there is a finite number of primes.Let G be the set of all primes [math]P_1, P_2, P_3... P_n[/math]Compute [math]K= P_1 \times P_2 \times P_3  \times ... \times P_n +1[/math].If K is prime, then it is obviously not in G.Otherwise, none of its prime factors are in G.Conclusion: G is not the set of all primes.For more detail one can look at thisEuclid's theorem2. There exist two irrational numbers x, y such that [math]x^y[/math] is rational. If [math]x=y=\sqrt{2}[/math] is an example, then we are done; otherwise [math]\sqrt{2}^\sqrt{2}[/math] is irrational, in which case taking [math]x=\sqrt{2}^\sqrt{2}[/math] and [math]y=\sqrt{2}[/math] gives us:[math]\left(\sqrt{2}^\sqrt{2} \right)^\sqrt{2} = \sqrt{2}^\left(\sqrt{2}\sqrt{2} \right) = \sqrt{2}^2 = 2[/math]Though now we have Gelfond-Schneider Theorem which implies this.Edit 1: One more proof which is a bit geometrically intuitive. In number theory, there is a curious relationship between the sum of consecutive cubes of the set of natural numbers and the square of the sum of the corresponding numbers themselves. It can be stated as: [math]\left(1 + 2 + 3 + ... +n \right)^2 = 1^3 + 2^3 +3^3 + ... + n^3 [/math]                       [math]n\varepsilon N[/math]​In the picture below I have made a square that is 15 by 15. The bars on the side and bottom of the square show that the square has an area equal to, 1 + 2 + 3 + 4 + 5. (Notice that I have colour coded the numbers). The total area of the square is equal to the sum of these numbers, squared (The left hand side of the problem.). I have arranged the squares of these numbers in the large square. This leaves an area to account for.​​In the next picture you will see that I have divided the remainder into squares. The two rectangles can be rearranged into squares.​​The total area can be written as shown.​This is the right hand side of the problem and completes the solution.This, of course is interesting because the solution requires an area model, when we would expect an volume problem.Source: page has some more interesting problems you can look at.

If you dig a hole through Earth ( south to North pole ) and you throw a ball in it , what will happen to it?

It would come back and hit you in your head hoping to knock some sense into you.