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Mathematics and Statistics for Financial Risk Management by Michael B. Miller

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CHAPTER 2

Probabilities

In this chapter we will explore the application of probabilities to risk management. We will also introduce basic terminology and notations that will be used throughout the rest of this book.

DISCRETE RANDOM VARIABLES

The concept of probability is central to risk management. Many concepts associated with probability are deceptively simple. The basics are easy, but there are many potential pitfalls.

In this chapter, we will be working with both discrete and continuous random variables. Discrete random variables can take on only a countable number of values—for example, a coin, which can only be heads or tails, or a bond, which can only have one of several letter ratings (AAA, AA, A, BBB, etc.). Assume we have a discrete random variable X, which can take various values, xi. Further assume that the probability of any given xi occurring is pi. We write:

(2.1) Numbered Display Equation

where P is our probability operator.1

An important property of a random variable is that the sum of all the probabilities must equal one. In other words, the probability of any event occurring must equal one. Something has to happen. Using our current notation, we have:

(2.2) Numbered Display Equation

CONTINUOUS RANDOM VARIABLES ...

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