Gaussian Distributions and Random Variables

Abstract

Assume that a, σ are real numbers, σ › 0. The Gaussiant^† distribution N(a, σ²) is a measure defined on the Borel σ-algebra B ¹ of the space ℝ¹, whose density with respect to the Lebesgue measure is

$$p(r) = {(2\pi )^{ - 1/2}}{\sigma ^2}\exp \{ - {(r - a)^2}/2{\sigma ^2}\} $$

(1)

.