Category: Discrete Random Variables

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…

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…

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,…

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…

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…

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…

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…

We start our investigation of random variables. Descriptive statistics deals with variables that can take values within a discrete or a continuous set. Correspondingly, we cover discrete random variables. As we shall see, the mathematics involved in the study of continuous random variables requires concepts from calculus and is a bit more challenging than what…