basic probability jean walrand eecs – u.c. berkeley
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Basic Probability
Jean Walrand
EECS – U.C. Berkeley
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Outline
1. Interpretation2. Probability Space3. Independence4. Bayes5. Random Variable6. Random Variables7. Expectation8. Conditional Expectation9. Notes10. References
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1. Interpretation
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2. Probability Space2.1. Finite Case
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2. Probability Space2.2. General Case
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2. Probability Space
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3. Independence
Each element has p = 1/4A B
C
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4. Bayes’ Rule
B1
B2
A
p1
p2
q1
q2
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4. Bayes’ RuleExample:
H0
H1
A = {X > 0.8}
p0
p1
q0
q1
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5. Random Variable
x
x0
1
0 1
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5. Random Variable
0.5 10.30x
FX(x)
0.210.31
0.650.45
1
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5. Random Variable
Slope = afX = 1
a
100
fY = 1/a
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5. Random Variable
Other examples:•Bernoulli•Binomial•Geometric•Poisson•Uniform•Exponential•Gaussian
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6. Random Variables
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6. Random VariablesExample 1
10
Uniform in triangle
X()
Y()
1
0
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6. Random VariablesExample 2
xy
g(.)x + dx y + H(x)dx
Scaling by |H(x)|
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7. Expectation
0.5 10.30x
FX(x)
0.210.31
0.650.45
1
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7. Expectation
Example:
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8. Conditional Expectation
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8. Conditional Expectation
X
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9. Notes Dependence ≠ Causality Pairwise ≠ Mutual Independence Random variable = (deterministic) function Random vector = collection of RVs Joint pdf is more than marginals E[X|Y] exists even if cond. density does not Most functions are Borel-measurable Easy to find X() that is not a RV Importance of prior in Bayes’ Rule. (Are you Bayesian?) Don’t be confused by mixed RVs
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10. Reference
Probability and Random Processes