## What is Normal Distribution?

In risk management, normal distribution is most widely employed. It’s also known as a bell curve or a Gaussian distribution (after mathematician Carl Friedrich Gauss) (which reflects the shape of the PDF).

Normal distribution is popular for many reasons.

- Many continuous random variables are approximately normally distributed.
- The distribution of many discrete random variables can be well approximated by a normal.
- The normal distribution plays a crucial role in the Central Limit Theorem (CLT), widely used in hypothesis testing.

Normal random variables are infinitely divisible, making them suitable for simulating asset prices in models that assume that prices are continuously evolving.

The normal distribution has two parameters: i.e. the mean and 2(the variance)

Therefore,

$ V[Y] = \sigma ^{2} $

and

$ E[Y] = \mu $

### PDF of Normal Distribution:

$ f_{Y}(y) = \frac{1}{\sqrt{2\Pi \sigma ^{2}}}exp(-\frac{(y-\mu )^{2}}{2\sigma ^{2}}) $

### CDF of Normal Distribution:

CDF of a normal does not have a closed-form; fast numerical approximations are widely available in Excel or other statistical software packages.

## Why is Normal Distribution important?

The higher the standard deviation, the riskier the investment, leading to more uncertainty. Hence, the graphical representation of normal distribution through its mean and standard deviation enables the representation of both returns and risk within a clearly defined range.

The normal is widely applied, and key quantiles from the normal distribution are commonly used for two purposes.

1. First, the quantiles are used to approximate the chance of observing values more than 1er, 2a, and 3a when describing a log return as a normal random variable.

2. Second, when constructing confidence intervals for estimated parameters, it is common to consider symmetric intervals that contain 90%, 95%, or 99% of the probability.