# The Gaussian Distribution

The Gaussian (Normal) distribution is a probability density function specified by two parameters – mean and variance . If a single variable, say , is normally distributed the density function is abbreviated as . The square root of the variance is called the standard deviation . The inverse of the variance is called the precision . The function is defined as follows

## Visualization

95% of the area under the Gaussian distribution curve lies within 2 standard deviation units about the mean.

## Positive

Because of the exponential term, the above density function is always positive.

## Maximum

The peak of the Gaussian distribution occurs when

## Normalized

Over the entire feature space, the density function integrates to 1, thus making it a valid probability distribution.

Define

Substituting and using the fact that is an even function

There are several tricks to solve this definite Gaussian integral by using the following property

Define

## Mean

The mean is defined by the expected value of the input variable over the entire feature space

Define

Substituting

The first term integrates to 0 because the integrand is an odd function and the integral is over the entire feature space. The integrand in the second term is another (zero-mean) gaussian distribution and as such integrates to 1 over the entire feature space. Thus

## Variance

The variance is defined by the expected squared deviation of the input variable from the mean over the entire feature space.

Define

Substituting

Using integration by parts

The first term integrates to 0 because the integrand is an odd function and the integral is over the entire feature space. The second term is the Gaussian integral which evaluates to . Thus