Measurements collected to represents height, weight and temperature fall under the classification of Continuous Random Variables. Many such continuous variables have bell shaped distributions and are know as Normal Random variables. Theoretically the distribution of a normal random variable is continuous, bell shaped and symmetric about the mean with median and mean coinciding with mean.

The equation for a normal curve is y = $\frac{1}{\sigma \sqrt{2\pi }}e^{\frac{-(x-\mu )^{2}}{2\sigma ^{2}}}$.where μ and σ represent the mean and standard deviation of the random variable which describe a normal distribution completely. The notation used to denote a normal distribution is N(μ, σ2) where σ is the variance of the distribution.

The probabilities of normal random variables are proportional to the area under the normal curve between the two end points that limit the variable. The theoretical normal curve can also be used to study random variable which are only approximately normal.

## Standard Normal Random Variable

To enable the study of all normal random variables easy using a single area table, standard normal variables are used. The standard normal distribution has mean μ =0 and standard deviation σ =1.
The formula for standard normal distribution is
y = $\frac{e^{-\frac{z^{2}}{2}}}{\sqrt{2\pi }}$ where z is called the standard normal variable.

The formula used to transform a normal random variable X to a standard normal variable z is
z = $\frac{X-\mu }{\sigma }$ where μ and σ are the mean and standard deviation of the normal random variable X.

## Standard Normal Random Variable Table

Standard normal (Z) table are used to calculate the areas under the standard normal curve, which represent the probabilities of the normal random variables under study. Two types of standard normal table are used;

Positive z-score table
This table contains areas corresponding to positive z values. The are found from the table is the area between the mean and a positive z score.  Using the symmetrical property of the normal curve, this table can also be used to calculate the area between the mean and a negative z-score.

Left Area z-score table
This table gives the area under the normal curve to the left of a calculated z value.

Standard Normal Random Variable Table is given below

## Linear Combination of Normal Random Variables

Suppose X1, X1, ......Xn are n mutually independent normal random variables with means μ1, μ2,......μn and variances σ12, σ22.......σn2.

Then their linear combination Y = $\sum_{i=1}^{n}c_{i}X_{i}$ is also a random variable with
mean = $\sum_{i=1}^{n}c_{i}\mu _{i}$ and variance = $\sum_{i=1}^{n}c_{i}^{2}\sigma _{i}^{2}$

## Product of Normal Random Variables

Even though the sum of two independent normal random variables X and Y is also a normal random variable, it is not true in the case of the product.

The product of two normal random variables X and Y both with mean 0 and variances σ12, σ22 written as PXY(u) can be shown as a Bessel function using advanced techniques.