# Normal Distribution

### Normal Random Variable

A continuous random variable, **X**, follows a normal distribution with mean, **μ**, and variance, **σ²**, and is denoted by **N(μ, σ²)**. It also must be noted:

1. The variable can take any value: (−∞, +∞)

2.The density function is the expression in terms of the mathematical equation of the Gaussian curve:

#### The Normal Curve

Domain: (−∞, +∞).

It is symmetric about the mean μ.

It has a maximum in the mean μ.

It's increasing to the mean μ and descending from it.

At points μ − σ and μ + σ it has inflection points.

The horizontal axis is an asymptote of the curve.

The **area** determined by the function and the x-axis is equal to **1**.

Since it is symmetrical about the axis x = µ, it leaves equal areas on either side of 0.5.

The probability is equivalent to the area under the curve.

p(μ − σ < X ≤ μ + σ) = 0.6826 = 68.26 %

p(μ − 2σ < X ≤ μ + 2σ) = 0.954 = 95.4 %

p(μ − 3σ < X ≤ μ + 3σ) = 0.997 = 99.7 %