RANDOM VARIABLES

In probability theory we work with events. The questions we may ask about events are quite limited, as they can either occur or not, and we may just investigate the probability of an event. In business management, more often than not we are interested in questions with a more quantitative twist, since events are linked to numerical values.

Example 6.1 Consider an airline adopting an overbooking strategy. The rationale behind overbooking is that aircraft capacity cannot be stored: If an aircraft with 50 seats flies with 10 empty ones, this does not mean that its capacity will be 60 on the next flight. Hence, many airlines accept bookings in excess of the actual capacity, in order to compensate for no-shows, i.e., passengers with a reservation, who do not show up at check-in. In real life, different tariffs are offered, defining the possibility of canceling the reservation and moving freely to another flight. In a simplified setting, the overbooking problem consists of determining the total number of bookings that we should accept. This decision does have an impact on the probability of an overbooking event, which occurs if any passenger cannot be accommodated at check-in. However, we are likely to be interested also in the cost of an overbooking strategy, as an overbooked passenger must be protected in some way, either by rerouting her to another flight or by offering overnight accommodation. Hence, we should associate numerical values with events.

Formally, a random variable is a mapping from events to numerical values.

DEFINITION 6.1 (Random variable) A random variable is a function mapping outcomes within a sample space Ω, to real numbers. This is sometimes denoted as follows:

We can also use the notation X(ω), with ω ∈ Ω, to emphasize the dependence on random outcomes.

In this definition, we should note that random variables are typically denoted by uppercase letters like X, whereas lowercase letters such as x are reserved to denote realizations of random variables, i.e., numerical values x = X(ω) corresponding to a specific event. Sometimes, an alternative notation is used, where  refers to a random variable and ∈ refers to a realizations; this is handy with Greek letters, as you may imagine, and it is common in economics. The definition above is actually a bit informal and definitely incomplete. We should keep in mind that probability measures are associated with events, i.e., sets of elementary outcomes. Hence, the definition of a random variable depends on the family F of events E ⊆ Ω. Hence, we should define a random variable referring to a probability space, consisting of the sample space Ω, the family of event F, and the probability measure P(E), for events E ∈ F. This may look a bit too abstract, but it is what we need to express information properly. For all of the practical purposes of this book, we need not be bothered about such technicalities, which are just outlined in Section 7.10 for the interested reader. All we need to know is that random experiments may result in numerical values that are practically relevant to us.

Example 6.2 Consider a lottery based on coin flipping, in which you win an amount €20 if the coin lands head, and you lose €10 otherwise. Then, we have a random variable X, such that

which readily translates to the probability distribution:

Given our little lottery above, typical questions we may ask are

• If we keep repeating the lottery, what is the average win on the long term?
• What about its variability?
• More generally, how can we measure the risk of a lottery?

Considering the first two questions, we are naturally reminded of concepts like mean and variance, that we encountered in descriptive statistics. Indeed, we show later how these concepts can be adapted to random variables. However, there is a fundamental twofold gap when we move from descriptive statistics to probability theory and random variables.

1. Descriptive statistics is basically backward-looking: We calculate summary measure based on past observations, which could be referred to a whole population or to a possibly small sample. On the contrary, probability theory is concerned with what may happen in the future. Hence, probability theory is intrinsically forward-looking.
2. Furthermore, we implicitly assume that we know everything we need about a random variable, i.e., we assume that we are endowed with complete knowledge about the uncertainty of the phenomenon that we are representing.¹ In a sense, probability theory assumes knowledge about whole populations, not samples.

Clearly, there is a link, as one could use a sample to learn something about probabilities of future events, but we have already seen in Section 5.1 that this frequentist view is only one possibility. This is quite relevant when we consider the third question in the bullet list above. When dealing with risk management, one should ask if the past history contains all of the possible events. To get the point, imagine collecting statistics about credit crunches due to mortgage-backed securities before 2007.

We assume that we know the whole set of values that a random variable may take, as well as their probabilities. This is the support of a discrete probability distribution.

DEFINITION 6.2 (Support of a discrete distribution) The support of a discrete probability distribution is the set of values , i = 1, 2, 3,… that the random variable X may take. We always assume that values in the support have been sorted, i.e., x_i ≤ x_i+1

As we see from this definition, the values x_i may be real numbers in general; quite often, however, they are restricted to nonnegative integer numbers, i.e., . Yet, it is important to understand that this is not the feature that makes this distribution discrete. In the lottery above, in principle, the payoffs could be real numbers like 3π and −27π. The point is that in a discrete distribution the support is either a finite or a countably infinite set of values.² In the former case, the support is a set of numbers , i = 1, 2,…, n. Having said that, in the following the support of discrete distributions will mostly consist of integer numbers, which may include zero or not, depending on the application, but it is useful to keep the general framework in mind.