1. University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 11: Probability and Statistics (Part 1)

2. University of Colorado Boulder  Lecture Quiz 4 Due Friday @ 5pm ◦ Due by 5pm on Friday  Homework 4 Due September 25 ◦ As mentioned in e-mail, it is different from one originally posted at the start of the semester  Final Make-up Lecture Today @ 4pm 2

3. University of Colorado Boulder  Axioms of Probability  Probability Distributions  Multivariate Distributions 3

4. University of Colorado Boulder 4 Axioms of Probability

5. University of Colorado Boulder  X is a random variable (RV) with a prescribed domain.  x is a realization of that variable.  Example: ◦ 0 < X < 1 ◦ x 1 = 0.232 ◦ x 2 = 0.854 ◦ x 3 = 0.055 ◦ etc 5

6. University of Colorado Boulder  The conceptual definition holds for a discrete distribution  Requires more mathematical rigor for a continuous distribution (more later) 6

7. University of Colorado Boulder  Probability of some event A occurring:  Probability of events A and B occurring:  Axioms: 7

8. University of Colorado Boulder 8

9. University of Colorado Boulder  Although we often see a probability written as a percentage, a true mathematical probability is a likelihood ratio 9

10. University of Colorado Boulder  Mathematical definition of conditional prob.: 10  Example:

11. University of Colorado Boulder  Two events are independent iff 11  Why is the latter true if A and B are independent?

12. University of Colorado Boulder 12 Probability Distributions

13. University of Colorado Boulder  Random variables are either: ◦ Discrete (exact values in a specified list) ◦ Continuous (any value in interval or intervals)  Examples of each: ◦ Discrete: ◦ Continuous: 13

14. University of Colorado Boulder  DRVs provide an easier entry to probability 14  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!

15. University of Colorado Boulder  Probability of X in [x,x+dx]: 15 where f(x) is the probability density function (PDF)  For CRVs, the probability axioms become:

16. University of Colorado Boulder 16 Using axiom 2 as a guide, how would we derive k in the following:

17. University of Colorado Boulder  For the cases X ≤ x, let F(x) be the cumulative distribution function (CDF) 17  It then follows that: ?? ?

18. University of Colorado Boulder 18 From the definition of the density and distribution functions we have: From axioms 1 and 2, we find:

19. University of Colorado Boulder 19 Multivariate Distributions

20. University of Colorado Boulder  The PDF for two RVs may be written as: 20  Hence, for two RVs:

21. University of Colorado Boulder  How do we compute probabilities given a multivariate PDF? 21

22. University of Colorado Boulder  We often want to examine probability behavior of one variable when given a multivariate distribution, i.e., 22 Marginal density fcn of X

23. University of Colorado Boulder  What would be the marginal probability density function of Y? 23

24. University of Colorado Boulder  What if I only care about the probability of one variable? 24  Alternatively,

25. University of Colorado Boulder  Analogous to definition previously discussed, but rooted in the PDFs and marginal distributions 25

26. University of Colorado Boulder  If X and Y are independent, then 26 ?? ?