Category: Continuous Random Variables

Successful investing in stock shares is typically deemed a risky and complex endeavor. However, the following piece of advice seems to offer a viable solution:26 Buy a stock. If its price goes up, sell it. If it goes down, don’t buy it. In this section we dig a little deeper into concepts related to measurability…

So far, we have considered a single random variable. However, more often than not, we have to deal with multiple random variables. There are two cases in which we have to do so: In practice, we may also have the two views in combination, i.e., multiple variables observed over a timespan of several periods. In…

Most financial investments entail some degree of risk. Imagine a bank holding a portfolio of assets; the bank should set aside enough capital to make up for possible losses on the portfolio. To determine how much capital the bank should hold, precise guidelines have been proposed, e.g., by the Basel committee. Risk measures play a…

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…