ASEN 5070: Statistical Orbit Determination I Fall 2013 Professor Brandon A. Jones

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…