Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach

CHAPTER 19

CONTINUOUS DISTRIBUTIONS

Unlike the discrete distributions described in the previous chapter, continuous distributions are used to describe random variables that can take any number of values. In this chapter, we present some common continuous distributions and their properties.

19.1 Basic Concepts and Facts

Definition 19.1 (Univariate Normal Distribution), A random variable X on (R, ) is said to be normally distributed with mean μ and standard deviation σ, written as X ~ N(μ, σ²), if and only if its probability density function is given by

(19.1)

A normal random variable is called a standard normal random variable if it has mean 0 and standard deviation 1. The probability density function of a standard normal random variable is denoted by φ(x).

The cumulative density function of a normal random variable with mean μ and standard deviation σ is given by

(19.2)

The cumulative density function of a standard normal random variable is denoted by Φ(x) or N(x).

Definition 19.2 (Lognormal Distribution). Let μ R and σ > 0. A random variable X on is said to have a lognormal distribution ...