Month: February 2023

A detailed coverage of multivariate distributions is beyond the scope but we should at least consider a generalization of normal distribution. A univariate normal distribution is characterized by its expected value μ and by its variance σ2. In the multivariate case, we have a vector of expected values μ and a covariance matrix Σ. We consider a random vector taking values in :… Continue reading JOINTLY NORMAL VARIABLES

The covariance is a generalization of variance. Hence, it is not surprising that it shares a relevant shortcoming: Its value depends on the unit of measurement of the underlying quantities. We recall that it is impossible to say whether a variance of 10,000 is large or small; a similar consideration applies to standard deviation, which… Continue reading The correlation coefficient

In Section 7.7 we dealt with sums of random variables, under the restrictive assumption of independence. Finally, armed with covariance, we may tackle the general case. THEOREM 8.4 (Variance of the sum/difference of two random variables) Given two random variables X and Y, the variance of their sum and difference is respectively. This theorem is somewhat reassuring; if… Continue reading Sums of random variables

If two random variables are not independent, it is natural to investigate their degree of dependence, which means finding a way to measure it and to take advantage of it. The second task leads to statistical modeling, which we will investigate later in the simplest case of linear regression. The first task is not as easy… Continue reading COVARIANCE AND CORRELATION

In the previous section we formally introduced the concept of the joint cumulative distribution function (CDF). In the case of two random variables, X and Y, this is a function FX,Y(x, y) of two arguments, giving the probability of the joint event {X ≤ x, Y ≤ y}: The joint CDF tells the whole story about how the two random variables are linked. Then,… Continue reading INDEPENDENT RANDOM VARIABLES

In order to fully appreciate the issues involved in characterizing the dependence of random variables, as well as to appreciate the role of independence, we should have some understanding of how to characterize the joint distribution of random variables.1 For the sake of simplicity, we will deal only with the case of two random variables with… Continue reading JOINT AND MARGINAL DISTRIBUTIONS

So far, when dealing with a sequence of random variables, we always assumed that they were independent. At last, we investigate the issue of dependence. To get the basic intuition, consider the hypothetical demand data in Table 8.1. Table 8.1 Demand data for two items: Are they independent? Can we say that the two random variables D1 and D2 are independent?… Continue reading Introduction