Probability and Statistics for Finance

Chapter 10. Continuous Probability Distributions

In this chapter, we introduce the concept of continuous probability distributions. We present the continuous distribution function with its corresponding density function, a function unique to continuous probability laws. In the chapter, parameters of location and scale such as the mean and higher moments—variance and skewness—are defined for the first time even though they will be discussed more thoroughly in Chapter 13.

The more commonly used distributions with appealing statistical properties that are used in finance will be presented in Chapter 11. In Chapter 12, we discuss the distributions that unlike the ones discussed in Chapter 11 are capable of dealing with extreme events.

CONTINUOUS PROBABILITY DISTRIBUTION DESCRIBED

Suppose we are interested in outcomes that are no longer countable. Examples of such outcomes in finance are daily logarithmic stock returns, bond yields, and exchange rates. Technically, without limitations caused by rounding to a certain number of digits, we could imagine that any real number could provide a feasible outcome for the daily logarithmic return of some stock. That is, the set of feasible values that the outcomes are drawn from (i.e., the space Ω) is uncountable. The events are described by continuous intervals such as, for example, (−0.05, 0.05], which, referring to our example with daily logarithmic returns, would represent the event that the return at a given observation is more than −5% and at ...