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Probability Theory

Probability Theory

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The theory of probability is a special case of general measure theory in mathematics and as such has its fundamental definitions and basic results derived from real analysis. In this chapter, we define the fundamentals of probability theory only to the extent necessary for its application to radar interferometry error analysis, while maintaining some fidelity to measure theory. The Cramer-Rao lower bound is introduced and will be used in later chapters to provide an estimate for unbiased angle-of-arrival estimates. The Weiss-Weinstein, Ziv-Zakai, and Bhattacharyya lower bounds for biased random processes are defined and applied in Chapter 7 to estimate the performance of angle-of-arrival in the presence of angle ambiguities.

Chapter Contents:

  • 2.1 Random Variable
  • 2.2 Probability Density
  • 2.3 Mean and Covariance
  • 2.4 Maximum Likelihood
  • 2.5 Cramer-Rao Lower Bound
  • 2.6 Lower Bounds for Biased Estimators
  • 2.6.1 Bhattacharyya Bound
  • 2.6.2 Bobrovsky-Zakai Bound
  • 2.6.3 Weiss-Weinstein Bound
  • 2.6.4 Ziv-Zakai Bound
  • References

Inspec keywords: probability; radar interferometry; error analysis

Other keywords: probability theory; radar interferometry error; Weiss-Weinstein lower bounds; general measure theory; angle ambiguities; Bhattacharyya lower bounds; Ziv-Zakai lower bounds; biased random processes; unbiased angle-of-arrival estimates; Cramer-Rao lower bound

Subjects: Other topics in statistics; Error analysis in numerical methods; Radar equipment, systems and applications

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