A random variable X is said to follow a Normal Distribution with parameters \mu and \sigma if it has a probability density function given by:

f(x;\mu,\sigma) =  \frac{1}{\sqrt{2\pi}\sigma}  e^{-\frac{1}{2\sigma^2} \left( x - \mu \right)^2} ; -\infty < x<\infty , -\infty < \mu <\infty , \sigma > 0

A random variable X is said to follow a Normal Distribution with parameters \mu and \sigma if it has a distribution function given by:

F(x) = \int_{-\infty}^{x}   \frac{1}{\sqrt{2\pi}\sigma}  e^{-\frac{1}{2\sigma^2} \left( y - \mu \right)^2} dy

 E(X) =   \int_{-\infty}^{\infty} x f(x;\mu,\sigma)dx \\
= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} (\mu + \sigma y) e^{-\frac{y^2}{2}}dy \left[ \textit{Putting, } \sigma y = x-\mu  \right] \\
= \mu + 0 \left[ \textit{Since, } ye^{-\frac{y^2}{2}}  \textit{is an odd function} \right] \\
\mu_2 = V(X) = E(X-\mu)^2 =  \int_{-\infty}^{\infty} (x - \mu)^2 f(x;\mu,\sigma)dx \\
= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} ( \sigma y)^2 e^{-\frac{y^2}{2}}dy \left[ \textit{Putting, } \sigma y = x-\mu  \right] \\
=  \frac{\sigma^2}{\sqrt{2\pi}} \int_{-\infty}^{\infty} y^2 e^{-\frac{y^2}{2}}dy  \\
=  2 \frac{\sigma^2}{\sqrt{2\pi}} \int_{0}^{\infty} y^2 e^{-\frac{y^2}{2}}dy  \left[ \textit{Since, } y^2e^{-\frac{y^2}{2}}  \textit{is an even function} \right] \\
= \frac{2\sigma^2}{\sqrt{2\pi}}  \int_{0}^{\infty} \sqrt{2 p} e^{-p}dp \left[ \textit{Putting, } 2p = y^2   \right] \\
= \frac{2 \sigma^2}{\sqrt{\pi}} \Gamma(3/2) = \sigma^2 \\
\mu_3 = E(X-\mu)^3 = 0 \\
\mu_4 = E(X-\mu)^4 =  \frac{4 \sigma^4}{\sqrt{\pi}} \Gamma(5/2) = 3 \sigma^4

Skewness and Kurtosis

\gamma_1 = \frac{\mu_3}{\mu_2^{3/2}} = 0 , \\
\gamma_2 = \frac{\mu_4}{\mu_2^2} -3 = 3-3 =0

Some Interesting Properties of Normal Distribution

If a continuous random variable X follows N(\mu,\sigma) , then

1. The distribution is symmetric about \mu .
2. The normal probability curve has point of inflection at \mu \pm \sigma .
3. Owing to the fact that a change in the mean value changes the location of the probability curve of a normal distribution and a change in the variance value changes the shape of the probability curve, \mu \textit{and } \sigma^2 are called the location and scale parameters respectively.
4. The distribution is symmetric about \mu and hence the mean, median and mode of the distribution are the same. Also all odd order moments about the mean are zero.
5. The mean deviation about mean is \sqrt{\frac{2}{\pi}} \sigma
6. A normal distribution with mean 0 and variance 1 is known as standard/unit normal distribution. And the pdf and pdf of the standard normal distribution is usually denoted by \phi(.) \textit{and} \Phi(.) respectively. It satisfies the following relations,

1. \phi(x) = \phi(-x) , \forall x\in \mathbb{R} \quad \\
2. \Phi(x) = 1 - \Phi(x) , \forall  x\in \mathbb{R} \\
3. \Phi(0) = 0.5 \qquad \quad ...........

4. \quad 1-\Phi(x) \\
= \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}} \left\{ \frac{1}{x} - \frac{1}{x^3} + \frac{1.3}{x^5} - \frac{1.3.5}{x^7} + \dots + (-1)^k \frac{1.3\dots(2k-1)}{x^{2k+1}}\right\}