A Course on Statistics for Finance

Appendix B

Normal Distributions

This appendix relates to univariate and multivariate Normal distributions. Vectors and matrices, as discussed in Appendix A, are used here.

B.1 Some Results for Univariate Normal Distributions

B.1.1 Definitions

The family of Normal distributions is a two-parameter family. The parameters are the mean µ and the variance σ2 . The standard deviation is the square root of the variance. The standard Normal distribution is the Normal distribution with mean 0 and variance 1. Its p.d.f. is

$\varphi (z)=\frac{1}{\sqrt{2\pi }}{e}^{-z^2/2},-\infty <z<\infty .$

The c.d.f. is $\Phi (z)={\displaystyle {\int }_{-\infty }^z\varphi (z)dz=\mathrm{Pr}\{Z\le z\}}$, where Z is a random variable having the standard Normal distribution.

If a random variable Y has a Normal distribution with mean ...