Looking at Definition 6.3, the similarity with how the sample mean is calculated in descriptive statistics, based on relative frequencies, is obvious. However, there are a few differences that we must always keep in mind. This is why it is definitely advisable to avoid the term “mean” altogether, when referring to random variables. Using the term “expected value” may be tiresome at first, but it will enhance clarity of thinking, which will pay off later, when dealing with inferential statistics. Hence, it is useful to gain a thorough understanding of the differences between expected value in probability theory and mean in descriptive statistics.
- The first striking difference is that the expected value involves an infinite series, when the probability distribution has infinite support. This cannot happen in descriptive statistics, since what we observe is a bounded range of data. If the sum converges, it must be the case that very large values have a small probability. Allowing for the occurrence of an unlikely, but quite significant event is important in risk management.
- A related difference is that the probabilities need not come from empirical data. We will see in Section 6.5 that distributions may be obtained by conceptual random experiments driven by some underlying mechanism, which allows for an infinite set of outcome. We refer to such distributions as “theoretical” to set them apart from empirical distributions based on sampled data. A statistical mean can only be empirical.
- The expected value is a number. Given a probability distribution, the expected value is what it is. In descriptive statistics, the mean need not be a number. It will be a number only if the mean pertains to a population. If the mean comes from a random sample, then it will be random variable. Any time we sample, we get a different value.
Despite these remarkable differences, there is indeed a link between expected value and mean, which will become quite clear when dealing with inferential statistics; we may use a sample mean to estimate an expected value. We see that, in a sense, probability theory assumes complete knowledge about the underlying uncertainty, which can be equivalently encoded in the form of a PMF or CDF. Thus, in a sense, in probability theory we always work with the whole population.
In the following, we will learn to interpret the expected value in different ways, depending on our purpose:
- The expected value is a basic feature of a probability distribution, i.e., a location measure.
- The expected value can be regarded as a long-run average. This second interpretation is less obvious because, among other things, it assumes that the characteristics of a random process will be constant in the future. To really clarify what we mean, we should formally state the law of large numbers, specifying under which conditions this interpretation is sensible. Since the involved issues are not trivial, this is left to the advanced Section 9.8.
- The expected value can be interpreted as a forecast. This interpretation stresses the forward-looking nature of an expectation, as compared with the backward-looking nature of descriptive statistics. We clarify what makes a “good” forecast and what we need in order to transform forecasts into management decisions. From a practical perspective, we should always take into account the danger of building a forecast based on past history.4
Leave a Reply