Colorado Center for Astrodynamics Research The University of Colorado 1 STATISTICAL ORBIT DETERMINATION Probability and statistics review ASEN 5070 LECTURE 12 and 13 9/21/09

Colorado Center for Astrodynamics Research The University of Colorado 2 Given with and a, b, c, and e are constants. a)Determine the A matrix b)If what is ? Assume initial conditions, are given at.

Colorado Center for Astrodynamics Research The University of Colorado 3 Given with and a, b, c, and e are constants. a)If what is ? Assume initial conditions, are given at. Write as a 1 st order system

Colorado Center for Astrodynamics Research The University of Colorado 4 If the matrix is not of full rank, i.e. does not exist, can we make exist by the proper choice of a weighting matrix, ? 1.T or F 2.Justify your answer The rank of the product AB of two matrices is less than or equal to The rank of A and is less than or equal to the rank of B. Hence the answer to (1) is false.

Colorado Center for Astrodynamics Research The University of Colorado 5 Given range observations in the 2-D flat earth problem, i.e. 1.Assume all parameters except and are known. We can solve for both and from range measurements taken simultaneously from two well separated tracking stations. T or F? 2.Justify your answer in terms of the rank of.

Colorado Center for Astrodynamics Research The University of Colorado 6 y x Station 1 Station 2 x s1, y s1 x s2, y s2 y o = y o * Note: may lie anywhere on this line and modified to accommodate it. Spacecraft at t = t i Incorrect y o

Colorado Center for Astrodynamics Research The University of Colorado 7

Colorado Center for Astrodynamics Research The University of Colorado 8 Assume we had both range and range rate; can we now solve for ?

Colorado Center for Astrodynamics Research The University of Colorado 9 The differential equation is (choose all correct answers) 1.2 nd order and 2 nd degree 2.2 nd order and 1 st degree 3.linear 4.nonlinear

Colorado Center for Astrodynamics Research The University of Colorado 10 Probability and Statistics

Probability and Statistics Review 11 Copyright 2006 Axioms of Probability 2. p(S)=1, S is the certain event

Colorado Center for Astrodynamics Research The University of Colorado 12 Venn Diagram

Probability and Statistics Review 13 Copyright 2006 Axioms of Probability

Probability and Statistics Review 14 Copyright 2006 Probability Density & Distribution Functions

Probability and Statistics Review 15 Copyright 2006 Probability Density & Distribution Functions

Probability and Statistics Review 16 Copyright 2006 Probability Density & Distribution Functions Example:

Probability and Statistics Review 17 Copyright 2006 Expected Values

Probability and Statistics Review 18 Copyright 2006 Expected Values

Probability and Statistics Review 19 Copyright 2006 Expected Values

Probability and Statistics Review 20 Copyright 2006 The Gaussian or Normal Density Function

Probability and Statistics Review 21 Copyright 2006 The Gaussian or Normal Density Function

Probability and Statistics Review 22 Copyright 2006

Probability and Statistics Review 23 Copyright 2006

Probability and Statistics Review 24 Copyright 2006

Probability and Statistics Review 25 Copyright 2006 Two Random Variables

Probability and Statistics Review 26 Copyright 2006 Marginal Distributions

Probability and Statistics Review 27 Copyright 2006 Marginal Distributions

Probability and Statistics Review 28 Copyright 2006 Independence of Random Variables

Probability and Statistics Review 29 Copyright 2006 Conditional Probability

Probability and Statistics Review 30 Copyright 2006 Expected Values of Bivariate Functions

Probability and Statistics Review 31 Copyright 2006 Expected Values of Bivariate Functions

Probability and Statistics Review 32 Copyright 2006

Colorado Center for Astrodynamics Research The University of Colorado 33 Example Problem Given Find: a)k b)The marginal density functions of x and y c)The probability that d) e) f)Whether x and y are independent g) h)

Colorado Center for Astrodynamics Research The University of Colorado 34 Example Problem a)

Colorado Center for Astrodynamics Research The University of Colorado 35 Example Problem b)

Colorado Center for Astrodynamics Research The University of Colorado 36 Example Problem c)

Colorado Center for Astrodynamics Research The University of Colorado 37 Example Problem d) also, as ranges from and ranges over and,

Colorado Center for Astrodynamics Research The University of Colorado 38 Example Problem e) f) Hence, they are not independent g)

Colorado Center for Astrodynamics Research The University of Colorado 39 Example Problem h) However, the probability that 0<y<1/2 has changed very little: why?

Probability and Statistics Review 40 Copyright 2006 The Variance-Covariance Matrix

Probability and Statistics Review 41 Copyright 2006 The Variance-Covariance Matrix

Probability and Statistics Review 42 Copyright 2006 Properties of the Correlation Coefficient

Probability and Statistics Review 43 Copyright 2006 Properties of Covariance and Correlation

Probability and Statistics Review 44 Copyright 2006 Properties of Covariance and Correlation

Colorado Center for Astrodynamics Research The University of Colorado 45 Example Problem continued Determine the Variance-Covariance matrix for the example problem elsewhere We have shown that the marginal density functions are given by

Colorado Center for Astrodynamics Research The University of Colorado 46 Example Problem continued The elements of the variance-covariance matrix are computed below

Colorado Center for Astrodynamics Research The University of Colorado 47 Example Problem continued Cont.

Colorado Center for Astrodynamics Research The University of Colorado 48 Example Problem continued The variance-covariance matrix, P, is given by The correlation coefficient for random variables x and y is given by The conventional OD expression for the variance-covariance matrix has variances on the diagonal, covariance's in the upper triangle and correlation coefficients in the lower triangle. Hence,

Probability and Statistics Review 49 Copyright 2006 Bivariate Normal Distribution

Probability and Statistics Review 50 Copyright 2006 Marginal Density Function

Probability and Statistics Review 51 Copyright 2006 Conditional Density Function

Probability and Statistics Review 52 Copyright 2006 Conditional Density Function

Probability and Statistics Review 53 Copyright 2006 Conditional Density Function

Probability and Statistics Review 54 Copyright 2006 The Multivariate Normal Distribution

Probability and Statistics Review 55 Copyright 2006 The Multivariate Normal Distribution

Probability and Statistics Review 56 Copyright 2006 Conditional Distribution for Multivariate Normal Variables

Probability and Statistics Review 57 Copyright 2006

Probability and Statistics Review 58 Copyright 2006 Central Limit Theorem

Probability and Statistics Review 59 Copyright 2006 Example: Central Limit Theorem

Probability and Statistics Review 60 Copyright 2006

Probability and Statistics Review 61 Copyright 2006 Central limit theorem