1. University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 14: Probability and Statistics (Part 4)

2. University of Colorado Boulder  Lecture Quiz Due by 5pm  Homework #5 Due 10/2  Exam 1 – Oct. 9 ◦ More details to come 2

3. University of Colorado Boulder  Variance-Covariance Matrix  Multivariate Gaussian Distribution  Central Limit Theorem  Bayes’ Theorem  Statistical Least Squares 3

4. University of Colorado Boulder 4 Variance-Covariance Matrix

5. University of Colorado Boulder  Covariance provides a measure of correlation between variables 5

6. University of Colorado Boulder 6  Indicates the degree of linear correlations between variables

7. University of Colorado Boulder  When we have a linear relationship between random variables, then we have an extreme value of the correlation coefficient, and vice versa  In other words,  See pages 458-459 of the textbook 7

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9. University of Colorado Boulder  Symmetric  Is it non-singular? 9

10. University of Colorado Boulder 10 Multivariate Normal Distribution

11. University of Colorado Boulder  Multivariate: 11  Univariate:

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13. University of Colorado Boulder  It may be shown that: 13  Although the above assumes a bivariate normal distribution, the idea extends to higher dimensions with minor changes

14. University of Colorado Boulder  The conditional density function is also a normal distribution (anyone seeing a trend here?)  What happens if ρ = 0?  What happens if ρ = ±1? 14

15. University of Colorado Boulder 15  The conditional PDF from the previous slide is a special case of the general conditional PDF

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17. University of Colorado Boulder 17 Central Limit Theorem

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20. University of Colorado Boulder  The CLT implies that we can treat ε as a Gaussian random variable  What about when we have a small number of observations from different sensors? 20

21. University of Colorado Boulder 21 Bayes’ Theorem

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23. University of Colorado Boulder  Allows for updating a hypothesis’ probability when given additional information ◦ Known as Bayesian Inference  Modern estimation research is rooted in Bayesian Inference! 23

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