Gaussian Random Variables (The Normal and the Multivariate Normal Distributions)

Abstract

Let us recall that a Normal random variable with parameters (μ, σ²), where μ ∈ R and σ² > 0, is a random variable whose density is given by:

$$ f\left( x \right) = \frac{1} {{\sqrt {2\pi \sigma } }}e^{{{ - \left( {x - \mu } \right)^2 } \mathord{\left/ {\vphantom {{ - \left( {x - \mu } \right)^2 } 2}} \right. \kern-\nulldelimiterspace} 2}\sigma ^2 } , - \infty < x < \infty . $$

(16.1)

Such a distribution is usually denoted N(μ, σ²). For convenience of notation, we extend the class of normal distributions to include the parameters μ ∈ R and σ² = 0 as follows: we will denote by N(μ, 0) the law of the constant r.v. equal to μ (this is also the dirac measure at point μ). Of course, the distribution N(μ, 0) has no density, and in this case we sometimes speak oF a degenerate normal distribution. When μ = 0 and σ² = 1, we say that N(0, 1) is the standard Normal distribution.