Category: Continuous Random Variables

A recurring task in applications is summing random variables. If we have n random variables Xi, i = 1,…, n, we may build another random variable What can we say about the distribution of Y? The answer depends on two important features of the terms in the sum: We will clarify what we mean by “independent random variables” formally but…

Sometimes, no theoretical distribution seems to fit available data, and we resort to an empirical distribution. A standard way to build an empirical distribution is based on order statistics, i.e., sorted values from a sample. Assume that we have a sample of n values and order statistics X(i), i =1,…, n, where X(i) ≤ X(i+1). The value X(1) is the smallest observation and X(n) is the largest one.…

The normal distribution is by far the most common, and misused, distribution in the theory of probability. It is also known as Gaussian distribution, but the term “normal” illustrates its central role quite aptly. Its PDF has a seemingly awkward form Fig. 7.14 PDF of two normal distributions. depending on two parameters, μ and σ2. Actually, we met such a…

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…

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…

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…

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…

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…

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…

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…