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… Continue reading PROBABILITY SPACES, MEASURABILITY, AND INFORMATION

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… Continue reading STOCHASTIC PROCESSES

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… Continue reading An application to finance: value at risk (VaR)

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… Continue reading Setting the reorder point in inventory control

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… Continue reading The newsvendor problem with normal demand

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… Continue reading MISCELLANEOUS APPLICATIONS

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… Continue reading The law of large numbers: a preview

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… Continue reading Central limit theorem

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… Continue reading Distributions obtained from the normal

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… Continue reading The square-root rule