Quantitative Methods: An Introduction for Business Management

7

Continuous Random Variables

In the previous chapter 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
• Last but not least, in the next chapters on statistical applications, the most common probability distribution ...