## Introduction

The Normal Distribution is also known as the *Gaussian Distribution *and *Bell-Shaped Distribution*. When a random experiment is replicated, the random variable that will equal the average or total result over the replicates will have a normal distribution as the number of replicates becomes large.

## Probability Density Function

For âˆ’âˆž<Î¼<âˆž and Ïƒ>0, the Normal Distribution is denoted by N(Î¼,Ïƒ2), and its probability density is given by

.

Ïƒ is the Standard Deviation, and Î¼ is the Mean.

Letâ€™s break the formula into smaller pieces to understand whatâ€™s happening.

**Z-score - **It measures how many standard deviations away a data point lies from the mean.

.

The exp in the above probability density formula is the square of the z-score times -Â½. The values that are away from the mean have a lower probability than those near the mean. The values away from the mean have a higher z-score and thus a lower probability since the exp is negative. The vice-versa is for the values closer to the mean.

Now comes the **68-95-99.7 rule. **The rule states that the % of values within a band around the mean in a normal distribution having a width of two, four, and six standard deviations comprise 68%, 95%, and 99.7% of all the values.

The effects of Ïƒ and Î¼ on the Distribution are shown in the above picture.

The above f(x) is integrated into one, which can be proved by the Gaussian integral below.

Let â€˜Xâ€™ = normal distributed random variable with parameters Î¼ and Ïƒ^2. The area inside the normal distribution curve is 1 as the probability is 1.

Thus, = 1;

writing x as (x-Î¼ ) + Î¼ yields

Letting y=x-Î¼

The first part is symmetric about the y-axis, hence its value is 0.

Thus,

Expectation E[x] = Î¼

Variance = Ïƒ^2

Standard Deviation = .