Month: February 2023

Say that we are in charge of managing the inventory of a component, whose supply lead time is 2 weeks. Weekly demand is modeled by a normal random variable with expected value 100 and standard deviation 20 (let us pretend that this makes sense). If we apply a reorder point policy based on the EOQ…

We know from Section 7.4.4 that the optimal solution of a newsvendor problem with continuous demand is the solution of the equation i.e., the quantile of demand distribution, corresponding to probability m/(m+cμ). If we assume normal demand, with expected value μ and standard deviation σ, then the optimal order quantity (assuming that we want to maximize expected profit) is Assume that…

In this section we outline a few applications from logistics and finance. The three examples will definitely look repetitive, and possibly boring, but this is exactly the point: Quantitative concepts may be applied to quite different situations, and this is why they are so valuable. In particular, we explore here three cases in which quantiles…

The sample mean plays a key role in descriptive statistics and, as we shall see, in inferential statistics as well. In this section we take a first step to characterize its properties and, in so doing, we begin to appreciate an often cited principle: the law of large numbers. Fig. 7.22 Histograms obtained by sampling the…

cAs we noted, it is difficult to tell which distribution we obtain when summing a few i.i.d. variables. Surprisingly, we can tell something pretty general when we sum a large number of such variables. We can get a clue by looking at Fig. 7.22. We see the histogram obtained by sampling the sum of independent exponential random variables…

As we pointed out, if we sum i.i.d. random variables, we may end up with a completely different distributions, with the normal as a notable exception. However, there are ways to combine independent normal random variables that lead to new distributions that have remarkable applications, among other things, in inferential statistics. In fact, statistical tables…

Consider a sequence of i.i.d. random variables observed over time, Xt, t = 1,…, T. Let μ and σ be the expected value and standard deviation of each Xt, respectively. Then, if we consider the sum over the T periods, , we have We see that the expected value scales linearly with time, whereas the standard deviation scales with the square root of time. Sometimes students and practitioners are…

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…