PROBABILITY THEORY:INDEPENDENT RANDOM VARIABLES AND RANDOM SAMPLES

By -
INDEPENDENT RANDOM VARIABLES AND RANDOM SAMPLES

The concept of independent events has already been defined; that is, E1 and E2 are inde- pendent events if, and only if,

INTRODUCTION TO OPERATIONS RESEARCH-0588

Thus, the independence of the random variables X1 and X2 implies that the joint CDF factors into the product of the CDF’s of the individual random variables. Furthermore, it is easily shown that if (X1,X2) is a discrete bivariate random variable, then X1 and X2 are independent random variables if, and only if, PX X (k, l) = PX (k)PX (l); in other words, P{X1 =

INTRODUCTION TO OPERATIONS RESEARCH-0589

for all s ant t. Thus, if X1, X2 are to be independent random variables, the joint density (or probability) function must factor into the product of the marginal density functions of the random variables. Using this result, it is easily seen that if X1, X2 are independent random variables, then the covariance of X1, X2 must be zero. Hence, the results on the variance of linear combinations of random variables given in Sec. 24.11 can be simpli- fied when the random variables are independent; that is,

INTRODUCTION TO OPERATIONS RESEARCH-0589

In other words, if X1 and X2 are independent, a knowledge of the outcome of one, say, X2, gives no information about the probability distribution of the other, say, X1. It was noted in the example in Sec. 24.10 on the time of first arrivals that the conditional den- sity of the arrival time of the first customer on the second day, given that the first cus- tomer on the first day arrived at time s, was equal to the marginal density of the arrival time of the first customer on the second day. Hence, X1 and X2 were independent random variables. In the example of the demand for a product during two consecutive months with the probabilities given in Sec. 24.9, it was seen in Sec. 24.10 that

INTRODUCTION TO OPERATIONS RESEARCH-0590Hence, the demands during each month were dependent (not independent) random variables.

The definition of independent random variables generally does not lend itself to de- termine whether or not random variables are independent in a probabilistic sense by look- ing at their outcomes. Instead, by analyzing the physical situation the experimenter usu- ally is able to make a judgment about whether the random variables are independent by ascertaining if the outcome of one will affect the probability distribution of the other.

The definition of independent random variables is easily extended to three or more random variables. For example, if the joint CDF of the n-dimensional random variable

INTRODUCTION TO OPERATIONS RESEARCH-0591

Comments

Post a Comment

Popular posts from this blog

DUALITY THEORY:THE ESSENCE OF DUALITY THEORY

By -
Image
THE ESSENCE OF DUALITY THEORY Given our standard form for the primal problem at the left (perhaps after conversion from another form), its dual problem has the form shown to the right. Thus, with the primal problem in maximizatio n form, the dual problem is in minimization form instead. Furthermore, the dual problem uses exactly the same pa r amete r s as the primal problem, but in different locations, as summarized below. 1. The coefficients in the objective function of the primal problem are the right-hand sides of the functional constraints in the dual problem. 2. The right-hand sides of the functional constraints in the primal problem are the coef- ficients in the objective function of the dual problem. 3. The coefficients of a variable in the functional constraints of the primal problem are the coefficients in a functional constraint of the dual problem. To highlight the comparison, now look at these same two problems in matrix notation (as introduced at the beginn
Read more

NETWORK OPTIMIZATION MODELS:THE MINIMUM SPANNING TREE PROBLEM

By -
Image
THE MINIMUM SPANNING TREE PROBLEM The minimum spanning tree problem bears some similarities to the main version of the shortest-path problem presented in the preceding section. In both cases, an undi r ecte d and connected network is being considered, where the given information includes some mea- sure of the positive length (distance, cost, time, etc.) associated with each link. Both problems also involve choosing a set of links that have the shortest total length among all sets of links that satisfy a certain property. For the shortest-path problem, this property is that the chosen links must provide a path between the origin and the destination. For the minimum spanning tree problem, the required property is that the chosen links must provide a path between ea c h pair of nodes. The minimum spanning tree problem can be summarized as follows: 1. You are given the nodes of a network but not the links. Instead, you are given the po- tential links and the positive length for each
Read more

NETWORK OPTIMIZATION MODELS:THE SHORTEST-PATH PROBLEM

By -
Image
THE SHORTEST-PATH PROBLEM Although several other versions of the shortest-path problem (including some for directed networks) are mentioned at the end of the section, we shall focus on the following simple version. Consider an undi r ecte d and connected network with two special nodes called the origin and the destination. Associated with each of the links (undirected arcs) is a non- negative distance. The objective is to find the shortest path (the path with the minimum total distance) from the origin to the destination. A relatively straightforward algorithm is available for this problem. The essence of this procedure is that it fans out from the origin, successively identifying the shortest path to each of the nodes of the network in the ascending order of their (shortest) distances from the origin, thereby solving the problem when the destination node is reached. We shall first outline the method and then illustrate it by solving the shortest-path problem encountered by the See
Read more