Handbook of Multi-Commodity Markets and Products: Structuring, Trading and Risk Management

APPENDIX A Quick Review of Distributions Relevant in Finance with Matlab^® Examples^*

Laura Ballotta and Gianluca Fusai

In this Appendix, we quickly review the properties of distributions relevant in finance, like the

• normal distribution
• lognormal distribution
• chi-square distribution
• non-central chi-square distribution
• Poisson distribution
• exponential distribution
• Gamma distribution
• multivariate Gaussian distribution.

We also present a standard procedure to simulate random variables via the so-called inverse method.

A.1 THE NORMAL DISTRIBUTION

Fact A.1.1 (Normal Distribution) A normal (Gaussian) random variable on ℜ with expected value μ ∈ ℜ and standard deviation σ ∈ ℜ⁺ has density function (pdf) ϕ_{μ, σ}(x) and cumulative distribution (cdf) Φ_{μ, σ}(x) given by:

and

We write . If μ = 0 and σ = 1 we have the so-called unit standard Gaussian random variable. In particular, if , then

Given the cdf, we can compute the probability that X falls in a given interval:

We produce