The Normal distribution is a family of Probability distributions that can be described by a two parameter density function given by:

f(x;μ,σ 2 )=exp(−(x−μ) 2 2σ 2 )2πσ 2 √ f(x;μ,σ2)=exp(−(x−μ)22σ2)2πσ2f(x;\mu, \sigma^2)=\frac {\exp\left({-\frac{(x-\mu)^2}{2\sigma^2}}\right)}{\sqrt{2\pi\sigma^2}}

The expected value of a random variable from this distribution is μ μ\mu and the variance is σ 2 σ2\sigma^2.

To answer the question of how it is derived, we need to make sense of what that question might mean. For example, if I tell you that a die is rolled n nn times and I ask for the derivation of the Probability mass function describing how many times I observe the number 5, that is a reasonably straightforward exercise.

I assume that you are asking for a similar "derivation" of the probability density function given above for the normal distribution. To begin such a derivation, we need to have a particular kind of random experiment in mind that should be well described by this distribution (analogous to the rolling of the die in the previous example). So what kinds of random processes should result in a normal distribution? It turns out that when the property that we are trying to model is the cumulative effect of a large number of independent (or weakly dependent) random effects all of which are typically approximately the same size, then the resulting random variable will follow a normal distribution.

For example, I would guess that the amount of rain that falls in my yard over the course of a year should be approximately normally distributed. Why? Because the total rainfall is the result of a large number of rain storms that take place throughout the year. The amount of rain that falls in a storm in October should be nearly independent of the amount of rain that falls in a storm in May. The amount of rain falling in each storm probably follows a similar distribution. So the combined effect of all such storms should be (roughly) normal.

OK, so how do we model a process like this mathematically in such a way that we arrive at the normal distribution. Here's the argument…

Let X i XiX_i for i∈N i∈Ni\in\mathbb N be a collection of independent, identically distributed random variables such that E(X i )=μ E(Xi)=μ\mathbb E(X_i)=\mu and Var(X i )=σ 2 <∞ Var(Xi)=σ2<∞\text{Var}(X_i)=\sigma^2<\infty. Let Z n =1n ∑ n i=1 X i −μ1n √ Zn=1n∑i=1nXi−μ1nZ_n=\frac{\frac 1n \sum_{i=1}^n X_i -\mu}{\sqrt{\frac{1}n}}. Then in the limit as n→∞ n→∞n\to\infty the density function of Z n ZnZ_n converges to f(x;0,σ 2 )=exp(−x 2 2σ 2 )2πσ 2 √ f(x;0,σ2)=exp(−x22σ2)2πσ2f(x;0, \sigma^2)=\frac {\exp\left({-\frac{x^2}{2\sigma^2}}\right)}{\sqrt{2\pi\sigma^2}}. This claim is (pretty much) a statement of the most basic form (i.e. the Lindeberg–Lévy version) of the Central limit theorem. The proof of this theorem is a little too substantial to give in an answer on Quora, but it is outlined here.

Answer by Michael Lamar:

The Normal distribution is a family of Probability distributions that can be described by a two parameter density function given by:[math]f(x;\mu, \sigma^2)=\frac {\exp\left({-\frac{(x-\mu)^2}{2\sigma^2}}\right)}{\sqrt{2\pi\sigma^2}}[/math]The expected value of a random variable from this distribution is [math]\mu[/math] and the variance is [math]\sigma^2[/math].To answer the question of how it is derived, we need to make sense of what that question might mean. For example, if I tell you that a die is rolled [math]n[/math] times and I ask for the derivation of the Probability mass function describing how many times I observe the number 5, that is a reasonably straightforward exercise.I assume that you are asking for a similar "derivation" of the probability density function given above for the normal distribution. To begin such a derivation, we need to have a particular kind of random experiment in mind that should be well described by this distribution (analogous to the rolling of the die in the previous example). So what kinds of random processes should result in a normal distribution? It turns out that when the property that we are trying to model is the cumulative effect of a large number of independent (or weakly dependent) random effects all of which are typically approximately the same size, then the resulting random variable will follow a normal distribution.For example, I would guess that the amount of rain that falls in my yard over the course of a year should be approximately normally distributed. Why? Because the total rainfall is the result of a large number of rain storms that take place throughout the year. The amount of rain that falls in a storm in October should be nearly independent of the amount of rain that falls in a storm in May. The amount of rain falling in each storm probably follows a similar distribution. So the combined effect of all such storms should be (roughly) normal.OK, so how do we model a process like this mathematically in such a way that we arrive at the normal distribution. Here's the argument…Let [math]X_i[/math] for [math]i\in\mathbb N[/math] be a collection of independent, identically distributed random variables such that [math]\mathbb E(X_i)=\mu[/math] and [math]\text{Var}(X_i)=\sigma^2<\infty[/math]. Let [math]Z_n=\frac{\frac 1n \sum_{i=1}^n X_i -\mu}{\sqrt{\frac{1}n}}[/math]. Then in the limit as [math]n\to\infty[/math] the density function of [math]Z_n[/math] converges to [math]f(x;0, \sigma^2)=\frac {\exp\left({-\frac{x^2}{2\sigma^2}}\right)}{\sqrt{2\pi\sigma^2}}[/math]. This claim is (pretty much) a statement of the most basic form (i.e. the Lindeberg–Lévy version) of the Central limit theorem. The proof of this theorem is a little too substantial to give in an answer on Quora, but it is outlined here.

What is a normal distribution curve? And How is it derived?