ASEN 5070: Statistical Orbit Determination I
Fall 2015
Professor Brandon A. Jones
Lecture 11: Probability and Statistics (Part 1)
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Lecture Quiz 4 Due Friday @ 5pm
◦ Due by 5pm on Friday
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Homework 4 Due September 25
◦ As mentioned in e-mail, it is different from one
originally posted at the start of the semester
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Final Make-up Lecture Today @ 4pm
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Axioms of Probability
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Probability Distributions
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Multivariate Distributions
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Axioms of Probability
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X is a random variable (RV) with a prescribed
domain.
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x is a realization of that variable.
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Example:
◦ 0<X<1
◦
◦
◦
◦
x1 = 0.232
x2 = 0.854
x3 = 0.055
etc
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The conceptual definition holds for a discrete
distribution
Requires more mathematical rigor for a
continuous distribution (more later)
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Probability of some event A occurring:
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Probability of events A and B occurring:
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Axioms:
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Although we often see a probability written as a
percentage, a true mathematical probability is a
likelihood ratio
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Mathematical definition of conditional prob.:
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Example:
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Two events are independent iff
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Why is the latter true if A and B are independent?
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Probability Distributions
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Random variables are either:
◦ Discrete (exact values in a specified list)
◦ Continuous (any value in interval or intervals)
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Examples of each:
◦ Discrete:
◦ Continuous:
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DRVs provide an easier entry to probability
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They are vary important to many aerospace
processes!
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However, StatOD tends to deal more with CRVs
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We will primarily discuss the latter!
◦ Rarely discretize the system of coordinates
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Probability of X in [x,x+dx]:
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where f(x) is the probability density function (PDF)
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For CRVs, the probability axioms become:
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Using axiom 2 as a guide, how would we derive k in the following:
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For the cases X ≤ x, let F(x) be the cumulative
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It then follows that:
distribution function (CDF)
??
?
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• From the definition of the density and distribution functions we have:
• From axioms 1 and 2, we find:
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Multivariate Distributions
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The PDF for two RVs may be written as:
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Hence, for two RVs:
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How do we compute probabilities given a
multivariate PDF?
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We often want to examine probability behavior of
one variable when given a multivariate
distribution, i.e.,
Marginal density fcn of X
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What would be the marginal probability
density function of Y?
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What if I only care about the probability of
one variable?
Alternatively,
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Analogous to definition previously discussed,
but rooted in the PDFs and marginal
distributions
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If X and Y are independent, then
???
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