Definition
Gaussian Distribution
The Gaussian distribution is a continuous distribution that is parameterised by
(mean) and (variance).
Properties
Affine Transform
Let
Standardisation
Let
Affine transformation to standard location and scale:
This works for any random variable
- Expectation:
- Variance:
.
Quantiles
Let
The
where
Symmetries
For the standard normal density, cdf, and
Standard
Definition
Link to originalStandard Gaussian Distribution
The standard Gaussian distribution is a Gaussian distribution with a mean
and a variance .
Probability Density Function
The probability density function is given by:
Parameter effects:
Central Limit Theorem
Definition
Link to originalCentral Limit Theorem
Let
be a sequence of identically distributed random variables with finite variance and
be the sample mean of the first random variables, then: Note that
is the first central moment of the sample’s distribution.