By Larry C. Andrews

Mathematical concept built in simple classes in engineering and technological know-how frequently consists of deterministic phenomena, and such is the case in fixing a differential equation that describes a few linear approach the place either the enter and output are deterministic amounts. In perform, notwithstanding, the enter to a linear approach, like an imaging method or radar method, may perhaps include a "random" volume that yields uncertainty concerning the output. Such structures needs to be handled through probabilistic equipment instead of deterministic tools. therefore, likelihood concept and random technique idea became critical instruments within the mathematical research of those sorts of engineering structures. subject matters incorporated during this SPIE box consultant are simple likelihood concept, random approaches, random fields, and random information research.

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**Extra resources for Field Guide to Probability, Random Processes, and Random Data Analysis**

**Example text**

Strictly speaking, stationarity is defined in terms of ensemble averages, but a slightly different interpretation takes place with a single time history of the process. Generally, one interprets the time history to be stationary if the properties computed over a short time interval do not change significantly from one interval to the next. A single stationary random process is generally described by the following statistical quantities: 1. mean and mean-square values 2. PDFs 3. correlation functions 4.

Field Guide to Probability, Random Processes, and Random Data Analysis Probability: Two Random Variables 25 Joint and Marginal Density Functions The joint density function of RVs x and y is defined by ∂2 Fxy ( x, y) f xy ( x, y) = ∂ x∂ y provided that the joint CDF Fxy ( x, y) is known, continuous, and differentiable. The joint PDF is also commonly called the bivariate PDF. , Fxy ( x, y) = x y −∞ −∞ f xy (ξ, η) d ξ d η If we define event A = {(x, y) ∈ D }, where ∈ means contained in and D is some domain of the ξη plane, then the probability assigned to this event is Pr( A ) = D f xy (ξ, η) d ξ d η The marginal CDF of RV x can be found from the joint CDF through the relation Fx ( x) = Fxy ( x, ∞) = x ∞ −∞ −∞ f xy (ξ, η) d ξ d η and by differentiating with respect to x, we obtain the marginal density function: f x ( x) = ∂ ∂x x ∞ −∞ −∞ f xy (ξ, η) d ξ d η = ∞ −∞ f xy ( x, η) d η Similarly, the marginal density function of y is f y ( y) = ∞ −∞ f xy (ξ, y) d ξ Field Guide to Probability, Random Processes, and Random Data Analysis 26 Probability: Two Random Variables Conditional Distributions and Density Functions To discuss conditional distributions, we illustrate only the case where the conditioning event depends on y.

It gives the best estimate of RV x, given the observation y = y. The maximum-likelihood estimate (MLE) xˆ is that value for which the conditional PDF f x ( x | y) is maximum. In statistics, the maximum-likelihood estimation (MLE) technique is a method of estimating the parameters of a statistical model. For a fixed set of data and underlying statistical model, the method of maximum likelihood selects values of the model parameters that give the observed data the greatest probability. , non-Gaussian) data.