Introduction

We have gained the essential intuition about random variables in the discrete setting. There, we introduced ways to characterize the distribution of a random variable by its PMF and CDF, as well as its expected value and variance. Now we move on to the more challenging case of a continuous random variable. There are several reasons for doing so:

• Some random variables are inherently continuous in nature. Consider the time elapsing between two successive occurrences of an event, like the request for service or a customer arrival to a facility. Time is a continuous quantity and, since this timespan cannot be negative, the support of a random variable modeling this kind of uncertainty is [0, +∞).
• Sometimes, continuous variables are used to model variables that are actually integers. As a practical example, consider demand for an item; a low-volume demand can be naturally modeled by a discrete random variable. However, when volumes are very high, it might be convenient to approximate demand by a continuous variable. To see the point, imagine a demand value like d = 2.7; in discrete manufacturing, you cannot sell 2.7 items, and rounding this value up and down makes a big difference; but what about d = 10,002.7? Quite often this turns out to be quite a convenient simplification, in both statistical modeling and in decision making.¹
• The most common probability distribution is, by far, the normal (or Gaussian) distribution, which is a continuous distribution whose support is the whole real line . As we will see, there are several reasons why the normal distribution plays a pivotal role in statistics.

The extends the concepts that we have introduced for the discrete case, and it also presents a few new ones that are better appreciated in this broader context. The mathematical machinery for a full appreciation of continuous random variables is definitely more challenging than that required in the discrete case. However, an intuitive approach is adequate to pursue applications to business management. Cutting a few corners, the essential difficulty in dealing with continuous random variables is that we cannot work with the probability P(X = x), as this is always zero for a continuous random variable. Unlike the discrete case, the probability mass is not concentrated at a discrete set of points, but it is distributed over a continuous set, which contains an infinite number of points even if the distribution support is a bounded interval like [a, b]. The role of the PMF is played here by a probability density function (PDF for short). Furthermore, the sums we have seen in the discrete context should be replaced by integrals.

Integrals were introduced in Section 2.13; we do not really need any in-depth knowledge, as integrals can be just interpreted as areas. In Section 7.1 we pursue an intuitive approach to see the link between such areas and probabilities. Then, we introduce density functions in Section 7.2, where we also see that the concept of cumulative distribution function (CDF) needs no adjustment when moving from discrete to continuous random variables. We see how expected values and variances are applied in this context in Section 7.3. Then, we expand our knowledge about the distribution of random variables by considering their mode, median, and quantiles in Section 7.4, and higher-order moments, skewness, and kurtosis in Section 7.5. All of these concepts apply to discrete random variables as well, but we have preferred to treat them once in the most general setting. As you can imagine, there is a remarkable variety of continuous distributions that can be applied in practice; we may use theoretical distributions whose parameters may be fit against empirical data, or we may just come up with an empirical distribution reflecting the data. In Section 7.6 we outline the main theoretical distributions – uniform, beta, triangular, exponential, and normal distributions – and we hint at how empirical distributions can be expressed. In Section 7.7 we take a first step toward statistical inference by considering sums of independent random variables; this will also lead us to the cornerstone central limit theorem, as well as a few more distributions that can be obtained from the normal and also play a pivotal role in inferential statistics; we will also have a preview of the often misunderstood law of large numbers. We illustrate a few applications in Section 7.8, with emphasis on quantiles of the normal distribution; what is remarkable, is that the very same concepts can be put to good use in diverse fields such as supply chain management and financial risk management. Finally, we consider sequences of random variables in time, i.e., stochastic processes, in Section 7.9. Section 7.10 can be skipped by most readers, as it has a more theoretical nature: Its aim is to clarify a point that we did not really investigate when we defined a random variable, i.e., the relationship between the random variable and the event structure of the underlying probability space.