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Introduction to Probability and Statistics in Statistical Orbit Determination

This lecture focuses on the foundational aspects of probability and statistics essential for statistical orbit determination. It covers key concepts such as random variables, probability distributions, and the axioms of probability. Understanding conditions for independence and calculating expected values are critical for analyzing data in aerospace processes. Multiple-choice quiz results demonstrate student comprehension, emphasizing the importance of both discrete and continuous random variables. The lecture serves as a primer for further exploration into multivariate distributions and their applications in statistical analysis.

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Introduction to Probability and Statistics in Statistical Orbit Determination

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  1. ASEN 5070: Statistical Orbit Determination I Fall 2013 Professor Brandon A. Jones Professor George H. Born Lecture 11: Probability and Statistics (Part 1)

  2. Announcements • Lecture Quiz Up • Due by 5pm on Wednesday • Homework 4 Due Friday • Material covered today and Wednesday

  3. Today’s Lecture • Lecture Quiz 2 • Axioms of Probability • Probability Distributions • Multivariate Distributions • Moment Generating Functions

  4. Lecture Quiz 2

  5. Question 1 • Percent Correct – 54.76%

  6. Question 2 • Percent Correct – 59.52%

  7. Question 3 • Percent Completely Correct – 54.76%

  8. Question 4

  9. Question 5 • Percent Correct – 90.48%

  10. Axioms of Probability

  11. Definitions and Symbols (for stats) • X is a random variable (RV) with a prescribed domain. • x is a realization of that variable. • Example: • 0 < X < 1 • x1 = 0.232 • x2 = 0.854 • x3 = 0.055 • etc

  12. Conceptual Definition • The conceptual definition holds for a discrete distribution • Requires more mathematical rigor for a continuous distribution (more later)

  13. Axioms of Probability • Probability of some event A occurring: • Probability of events A and B occurring: • Axioms:

  14. Illustration of Axioms 1 & 2

  15. Axioms of Probability • Although we often see a probability written as a percentage, a true mathematical probability is a likelihood ratio

  16. Conditional Probability • Mathematical definition of conditional prob.: • Example:

  17. Independence • Two events are independent iff • Why is the latter true if A and B are independent?

  18. Probability Distributions

  19. Random Variable Types • Random variables are either: • Discrete (exact values in a specified list) • Continuous (any value in interval or intervals) • Examples of each: • Discrete: • Continuous:

  20. Discrete Random Variables • DRVs provide an easier entry to probability • They are vary important to many aerospace processes! • However, StatOD tends to deal more with CRVs • Rarely discretize the system of coordinates • We will primarily discuss the latter!

  21. Continuous Probabilities • Probability of X in [x,x+dx]: where f(x) is the probability density function (PDF) • For CRVs, the probability axioms become:

  22. Implications of Axiom 2 Using axiom 2 as a guide, how would we derive k in the following:

  23. Distribution Functions • For the cases X ≤ x, let F(x) be the cumulative distribution function (CDF) • It then follows that: ???

  24. Example Continuous PDF • From the definition of the density and distribution functions we have: • From axioms 1 and 2, we find:

  25. Multivariate Distributions

  26. Multivariate Density Functions • The PDF for two RVs may be written as: • Hence, for two RVs:

  27. Multivariate Probabilities • How do we compute probabilities given a multivariate PDF?

  28. Marginal Distributions • We often want to examine probability behavior of one variable when given a multivariate distribution, i.e., Marginal density fcn of X

  29. Marginal Distributions • What would be the marginal distribution of Y?

  30. Probabilities of Only One Variable • What if I only care about the probability of one variable? • Alternatively,

  31. Independence of CRVs • Analogous to definition previously discussed, but rooted in the PDFs and marginal distributions

  32. Conditional Probabilities • If X and Y are independent, then ???

  33. PDF Moments

  34. Expected Value (mean) • The expected value is a weighted average of all possible values to determine the mean • Define the k-th moment about the origin as

  35. Moments about the Mean • We define the k-th moment about the mean: • The second moment about the mean is also known as the variance:

  36. Comparison of Mean and Variance

  37. Important Identity • Although not traditionally examined, higher-order moments are becoming increasingly important in orbit determination…

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