Month: February 2023

The exponential distribution is one the main tools used to model uncertainty, and it is related to other distributions, as well as to an important family of stochastic processes that we will investigate later. An exponential random variable can only take nonnegative values, i.e., its support is [0, +∞), and it owes its name to… Continue reading Exponential distribution

Triangular distribution is a possible model of uncertainty when limited knowledge is available. Three parameters characterize it: the extreme points of the support [a, b] and that the mode c, where a ≤ c ≤ b. The PDF for a triangular random variable is depicted in Fig. 7.11. The expected value and variance for a triangular distribution are respectively. Imagine a project planning… Continue reading Triangular and beta distributions

We have already met the uniform distribution in Section 7.1, where we specified its PDF and CDF. To say that a random variable X is uniformly distributed on the interval [a, b], the notation X ∼ U(a, b) is used. We have already shown that the expected value is the midpoint on the support: Since the uniform distribution is symmetric, the median… Continue reading Uniform distribution

In the following sections we describe some continuous probability distributions. The main criterion of classification is theoretical vs. empirical distributions. The former class consists of distributions that are characterized by a very few parameters; indeed, they can also be labeled as parametric distributions. Theoretical distributions will never fit empirical data exactly, but they provide us… Continue reading A FEW USEFUL CONTINUOUS PROBABILITY DISTRIBUTIONS

Expected value and variance do not tell us the whole story about a random variable. To begin with, they do not say anything about the possible lack of symmetry. From descriptive statistics, we know that to characterize symmetry of a distribution, or lack thereof, we need a coefficient measuring its skewness. Furthermore, we may have distributions… Continue reading HIGHER-ORDER MOMENTS, SKEWNESS, AND KURTOSIS

In Example 6.9 we have considered and solved numerically a hypothetical instance of the newsvendor problem. The procedure was based on brute force and did not provide us with any valuable insight into the structure of the problem itself. Furthermore, if we approximate the distribution of demand by a continuous distribution, which makes sense for high sale… Continue reading An application: the newsvendor problem again

Computing quantiles for a discrete random variable by applying Definition 7.1 would require inverting the CDF. However, this is a piecewise constant function, featuring jumps at each value of the distribution support, which makes its inversion impossible in general. Example 7.4 Consider random demand for a spare part, sold in low volumes, over the next time period. There… Continue reading Quantiles for discrete random variables

Roughly speaking, the median is a value splitting a dataset into two equal parts. When dealing with continuous random variables, we find that the median is a value mX such that Geometrically, the median splits the PDF in two parts with an area equal to 0.5. In descriptive statistics, the median can be regarded as a specific… Continue reading Median and quantiles for continuous random variables

The descriptive statistics, we have introduced concepts like mode, median, and percentiles. We have also remarked that some concepts, in particular the percentiles, are somewhat shaky in the sense that there are slightly different definitions and ways of calculating them using observed data. In this section we examine probabilistic counterparts of these concepts, and how… Continue reading MODE, MEDIAN, AND QUANTILES

Given a continuous random variable X and its PDF fX(x), its expected value is defined as follows: Quite often, we use the short-hand notation μX = E[X]. Again, this is straightforward extension of the discrete case, where E[X] ≡ ∑i xipX(xi). Example 7.2 As an illustration, let us consider the expected value of a uniform random variable on [a, b]. Symmetry suggests that… Continue reading EXPECTED VALUE AND VARIANCE