University of Colorado Boulder ASEN 5070 Statistical Orbit determination I Fall 2012 Professor George H. Born Professor Jeffrey S. Parker Lecture 5: Stat OD 1

University of Colorado Boulder  Homework 2 due Thursday  Homework 3 out today ◦ Basic dynamical systems relationships ◦ Studies of the state transition matrix ◦ Linear algebra  I’m unavailable this Wednesday. Use those TAs and is great of course. 2

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University of Colorado Boulder 6 ~N( 0.0, 1.41 )

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University of Colorado Boulder 8 ~N( 1.0, 0.41 )

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University of Colorado Boulder  Some popular questions and answers  Energy with Drag 11

University of Colorado Boulder  Some popular questions and answers  Computation of Time of Perigee 12

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University of Colorado Boulder  Chapter 4, Problems 1-6 ◦ Solving ODEs ◦ Linear Algebra ◦ Studying the state transition matrix 16

University of Colorado Boulder  Review of Differential Equations ◦ Laplace Transforms  Review of Statistics 17

University of Colorado Boulder  Stat OD dynamics:  Solve for given A and 18

University of Colorado Boulder  Stat OD dynamics:  Solve for given A and 19

University of Colorado Boulder  Solve for  w/ 20

University of Colorado Boulder  Solve for  w/ 21

University of Colorado Boulder  Solve the ODE  We can solve this using “traditional” calculus: 22 Check your answer by plugging it back in

University of Colorado Boulder  Laplace Transforms are useful for analysis of linear time-invariant systems: ◦ electrical circuits, ◦ harmonic oscillators, ◦ optical devices, ◦ mechanical systems, ◦ even orbit problems.  Transformation from the time domain into the frequency domain.  Inverse Laplace Transform converts the system back. 23

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University of Colorado Boulder  Solve the ODE  We can solve this using “traditional” calculus: 25

University of Colorado Boulder  Solve the ODE  Or, we can solve this using Laplace Transforms: 26

University of Colorado Boulder  Solve the ODE 27

University of Colorado Boulder  Solve the ODE 28

University of Colorado Boulder  Solve the ODE 29

University of Colorado Boulder  Solve the ODE 30

University of Colorado Boulder  Solve the ODE 31

University of Colorado Boulder  Solve the ODE 32

University of Colorado Boulder  Solve the ODE 33

University of Colorado Boulder  Questions on Diff EQ?  Quick Break  Review of Statistics to follow 34

University of Colorado Boulder  X is a random variable with a prescribed domain.  x is a realization of that variable.  Example: ◦ 0 < X < 1 ◦ x 1 = ◦ x 2 = ◦ x 3 = ◦ etc 35

University of Colorado Boulder Axioms of Probability 2. p(S)=1, S is the certain event

University of Colorado Boulder

University of Colorado Boulder Axioms of Probability

University of Colorado Boulder For the continuous random variable, axioms 1 and 2 become Probability Density & Distribution Functions

University of Colorado Boulder For the continuous random variable, axioms 1 and 2 become The third axiom becomes Which for a < b < c Probability Density & Distribution Functions

University of Colorado Boulder Probability Density & Distribution Functions Using axiom 2 as a guide, solve the following for k:

University of Colorado Boulder Probability Density & Distribution Functions Using axiom 2 as a guide, solve the following for k:

University of Colorado Boulder Probability Density & Distribution Functions

University of Colorado Boulder Example: From the definition of the density and distribution functions we have: From axioms 1 and 2, we find: Probability Density & Distribution Functions

University of Colorado Boulder Expected Values Note that:

University of Colorado Boulder Expected Values

University of Colorado Boulder Expected Values

University of Colorado Boulder Expected Values

University of Colorado Boulder Expected Values

University of Colorado Boulder The Gaussian or Normal Density Function

University of Colorado Boulder The Gaussian or Normal Density Function

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University of Colorado Boulder Two Random Variables

University of Colorado Boulder Marginal Distributions

University of Colorado Boulder Marginal Distributions

University of Colorado Boulder Independence of Random Variables

University of Colorado Boulder Conditional Probability

University of Colorado Boulder Expected Values of Bivariate Functions

University of Colorado Boulder Expected Values of Bivariate Functions

University of Colorado Boulder Expected Values of Bivariate Functions

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University of Colorado Boulder The Variance-Covariance Matrix

University of Colorado Boulder Properties of the Correlation Coefficient

University of Colorado Boulder Properties of Covariance and Correlation

University of Colorado Boulder Properties of Covariance and Correlation

University of Colorado Boulder Central Limit Theorem

University of Colorado Boulder  Addition of multiple variables taken from any single distribution  Gaussian  Example: Uniform [0,1] 70 ~N( 1.0, 0.41 ) ~N( 0.5, 0.29 ) ~N( 1.5, 0.50 ) ~N( 2.0, 0.58 ) Sum of 1 var Sum of 2 varsSum of 3 vars Sum of 4 vars

University of Colorado Boulder  Addition of multiple variables taken from any single distribution  Gaussian  Example: Uniform {0,1,2} (Quiz Question #4) 71 ~N( 2.0, 1.16 ) ~N( 1.0, 0.82 ) ~N( 3.0, 1.42 ) ~N( 10, 2.58 ) Sum of 1 var Sum of 2 varsSum of 3 vars Sum of 10 vars

University of Colorado Boulder  Addition of multiple variables taken from any single distribution  Gaussian  Example: Skewed distribution 72 ~N( 0.5, 0.40 ) ~N( 0.25, 0.28 ) ~N( 0.75, 0.49 ) ~N( 1.0, 0.57 ) Sum of 1 var Sum of 2 varsSum of 3 vars Sum of 4 vars

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University of Colorado Boulder  Questions on Statistics?  I’ll go through example problems at the beginning of Thursday’s lecture  Homework 2 due Thursday  Homework 3 out today  Next quiz active tomorrow at 1pm. 74