Probability and Statistics for Finance

Chapter 13. Parameters of Location and Scale of Random Variables

In the previous four chapters, we presented discrete and continuous probability distributions. It is common to summarize distributions by various measures. The most important of these measures are the parameters of location and scale. While some of these parameters have been mentioned in the context of certain probability distributions in the previous chapters, we introduce them here as well as additional ones.

In this chapter, we present as parameters of location quantiles, the mode, and the mean. The mean is introduced in the context of the moments of a distribution. Quantiles help in assessing where some random variable assumes values with a specified probability. In particular, such quantiles are given by the lower and upper quartiles as well as the median. In the context of portfolio risk, the so-called value-at-risk measure is used. As we will see, this measure is defined as the minimum loss some portfolio incurs with specified probability.

As parameters of scale, we introduce moments of higher order: variance together with the standard deviation, skewness, and kurtosis. The variance is the so-called second central moment and the related standard deviation are the most commonly used risk measures in the context of portfolio returns. However, their use can sometimes be misleading because, as was noted in Chapter 11, some distributions particularly suited to model financial asset returns have no finite variance. ...