Month: February 2023

The first thing we should observe is that variance cannot be negative, as it is the expected value of a squared deviation. It is zero for a random variable that is not random at all, i.e., a constant. In doing calculations, the following identity is quite useful: This is the analog of Eqs. (4.5) and (4.6) in… Continue reading Properties of variance

The expected value of a random variable tells us something about the location of its distribution, but we need a characterization of dispersion and risk as well. In descriptive statistics, we consider squared deviations with respect to the mean. Here we do basically the same thing, with respect to the expected value. DEFINITION 6.9 (Variance… Continue reading VARIANCE AND STANDARD DEVIATION

Typically, a random variable is just a risk factor that will affect some managerially more relevant outcome linked to cost or profit. This link may be represented by a function; hence, we are interested in functions of random variables. Given a random variable X and a function, like g(x) = x2, or g(x) = max{x, 0}, we define a new… Continue reading Expected value of a function of a random variable

We may think of the expected value as an operator mapping a random variable X into its expected value μ = E[X]. The expectation operator enjoys two very useful properties. PROPERTY 6.6 (Linearity of expectation 1) Given a random variable X with expected value E[X], we have for any numbers α and β. This property is fairly easy to prove: Informally, the… Continue reading Properties of expectation

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… Continue reading Expected value vs. mean

Both PMF and CDF provide us with all of the relevant information about a discrete random variable, maybe too much. In descriptive statistics, we use summary measures, such as mean, median, mode, variance, and standard deviation, to get a feeling for some essential features of a distribution, like its location and dispersion. In probability theory,… Continue reading EXPECTED VALUE

The CDF looks like a somewhat weird way of describing the distribution of a random variable. A more natural idea is just assigning a probability to each possible outcome in the support. Unfortunately, in the next chapter we will see that this idea cannot be applied to a continuous random variable. Nevertheless, in the case… Continue reading Probability mass function

The basic stuff of probability theory consists of events and their probability measures. Given a random variable X, consider the event {X ≤ x}; incidentally, note how we use x to denote a number. The probability of this event is a function of x. DEFINITION 6.3 (Cumulative distribution function) Let X be a random variable. The function for , is called cumulative distribution function (CDF). The notation FX(x) clarifies… Continue reading Cumulative distribution function

When we deal with sampled data, it is customary to plot a histogram of relative frequencies in order to figure out how the data are distributed. When we consider a discrete random variable, we may use more or less the same concepts in order to provide a full characterization of uncertainty. For instance, if we consider… Continue reading CHARACTERIZING DISCRETE DISTRIBUTIONS

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… Continue reading RANDOM VARIABLES