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ASEN 5070: Statistical Orbit Determination I Fall 2013 Professor Brandon A. Jones Professor George H. Born Lecture 11: Probability and Statistics (Part 1). Announcements. Lecture Quiz Up Due by 5pm on Wednesday Homework 4 Due Friday Material covered today and Wednesday.
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ASEN 5070: Statistical Orbit Determination I Fall 2013 Professor Brandon A. Jones Professor George H. Born Lecture 11: Probability and Statistics (Part 1)
Announcements • Lecture Quiz Up • Due by 5pm on Wednesday • Homework 4 Due Friday • Material covered today and Wednesday
Today’s Lecture • Lecture Quiz 2 • Axioms of Probability • Probability Distributions • Multivariate Distributions • Moment Generating Functions
Question 1 • Percent Correct – 54.76%
Question 2 • Percent Correct – 59.52%
Question 3 • Percent Completely Correct – 54.76%
Question 5 • Percent Correct – 90.48%
Definitions and Symbols (for stats) • X is a random variable (RV) 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
Conceptual Definition • The conceptual definition holds for a discrete distribution • Requires more mathematical rigor for a continuous distribution (more later)
Axioms of Probability • Probability of some event A occurring: • Probability of events A and B occurring: • Axioms:
Axioms of Probability • Although we often see a probability written as a percentage, a true mathematical probability is a likelihood ratio
Conditional Probability • Mathematical definition of conditional prob.: • Example:
Independence • Two events are independent iff • Why is the latter true if A and B are independent?
Random Variable Types • Random variables are either: • Discrete (exact values in a specified list) • Continuous (any value in interval or intervals) • Examples of each: • Discrete: • Continuous:
Discrete Random Variables • DRVs provide an easier entry to probability • 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!
Continuous Probabilities • Probability of X in [x,x+dx]: where f(x) is the probability density function (PDF) • For CRVs, the probability axioms become:
Implications of Axiom 2 Using axiom 2 as a guide, how would we derive k in the following:
Distribution Functions • For the cases X ≤ x, let F(x) be the cumulative distribution function (CDF) • It then follows that: ???
Example Continuous PDF • From the definition of the density and distribution functions we have: • From axioms 1 and 2, we find:
Multivariate Density Functions • The PDF for two RVs may be written as: • Hence, for two RVs:
Multivariate Probabilities • How do we compute probabilities given a multivariate PDF?
Marginal Distributions • We often want to examine probability behavior of one variable when given a multivariate distribution, i.e., Marginal density fcn of X
Marginal Distributions • What would be the marginal distribution of Y?
Probabilities of Only One Variable • What if I only care about the probability of one variable? • Alternatively,
Independence of CRVs • Analogous to definition previously discussed, but rooted in the PDFs and marginal distributions
Conditional Probabilities • If X and Y are independent, then ???
Expected Value (mean) • The expected value is a weighted average of all possible values to determine the mean • Define the k-th moment about the origin as
Moments about the Mean • We define the k-th moment about the mean: • The second moment about the mean is also known as the variance:
Important Identity • Although not traditionally examined, higher-order moments are becoming increasingly important in orbit determination…