Two-dimensional Distributions

In Chap. 5 is given basic information concerning two-dimensional distributions of random variables. Starting from a classical problem of the accuracy of artillery fire it is shown that, besides a traditional analytical procedure, components of a covariance tensor may be transformed by means of their representation by Mohr circles. It is shown that this representation, used commonly in mechanics of solids, for transformations of stress tensors, may be useful in solving such problems as a linear regression of experimental observations or a linear correlation between experimentally determined quantities. This method is compared with standard procedures. Examples of application illustrate advantages of using the representation of covariance tensors by Mohr circles. Theory of two-dimensional continuous distributions of random variables is described, with special attention given to a two-dimensional normal distribution along with a procedure of determining ellipses of probability concentration including information concerning a chi-squared distribution. It is shown how the normal two-dimensional distribution may be used for analysis of a gravity flow of granular media in bins and for analysis of a terrain subsidence caused by underground exploitation or by tectonic movements of a bedrock.