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ASEN 5070 Statistical Orbit determination I Fall 2012 Professor George H. Born

ASEN 5070 Statistical Orbit determination I Fall 2012 Professor George H. Born Professor Jeffrey S. Parker Lecture 5: Stat OD. Announcements. Homework 2 due Thursday Homework 3 out today Basic dynamical systems relationships Studies of the state transition matrix Linear algebra

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ASEN 5070 Statistical Orbit determination I Fall 2012 Professor George H. Born

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

  2. Announcements • 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 email is great of course.

  3. Quiz Results

  4. Quiz Results

  5. Quiz Results

  6. Quiz Results ~N( 0.0, 1.41 )

  7. Quiz Results

  8. Quiz Results ~N( 1.0, 0.41 )

  9. Quiz Results

  10. Quiz Results

  11. Homework 2 • Some popular questions and answers • Energy with Drag

  12. Homework 2 • Some popular questions and answers • Computation of Time of Perigee

  13. Homework 2: Tp Calculation

  14. Homework 2: Tp Calculation

  15. Homework 2: Tp Calculation

  16. Homework 3 • Chapter 4, Problems 1-6 • Solving ODEs • Linear Algebra • Studying the state transition matrix

  17. Today’s Lecture • Review of Differential Equations • Laplace Transforms • Review of Statistics

  18. Review of Diff EQ • Stat OD dynamics: • Solve for given A and

  19. Review of Diff EQ • Stat OD dynamics: • Solve for given A and

  20. Review of Diff EQ • Solve for • w/

  21. Review of Diff EQ • Solve for • w/

  22. Example • Solve the ODE • We can solve this using “traditional” calculus: Check your answer by plugging it back in

  23. Laplace Transforms • 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.

  24. Laplace Transform Tables

  25. Example • Solve the ODE • We can solve this using “traditional” calculus:

  26. Example • Solve the ODE • Or, we can solve this using Laplace Transforms:

  27. Applied to Stat OD • Solve the ODE

  28. Applied to Stat OD • Solve the ODE

  29. Applied to Stat OD • Solve the ODE

  30. Applied to Stat OD • Solve the ODE

  31. Applied to Stat OD • Solve the ODE

  32. Applied to Stat OD • Solve the ODE

  33. Applied to Stat OD • Solve the ODE

  34. Questions? • Questions on Diff EQ? • Quick Break • Review of Statistics to follow

  35. Review of Statistics • X is a random variable 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

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

  37. Venn Diagram

  38. Axioms of Probability

  39. Probability Density & Distribution Functions • For the continuous random variable, axioms 1 and 2 become

  40. Probability Density & Distribution Functions • For the continuous random variable, axioms 1 and 2 become • The third axiom becomes • Which for a < b < c

  41. Probability Density & Distribution Functions Using axiom 2 as a guide, solve the following for k:

  42. Probability Density & Distribution Functions Using axiom 2 as a guide, solve the following for k:

  43. Probability Density & Distribution Functions

  44. Probability Density & Distribution Functions Example: • From the definition of the density and distribution functions we have: • From axioms 1 and 2, we find:

  45. Expected Values Note that:

  46. Expected Values

  47. Expected Values

  48. Expected Values

  49. Expected Values

  50. The Gaussian or Normal Density Function

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