Basic Probability Jean Walrand EECS – U.C. Berkeley
Dec 19, 2015
Basic Probability
Jean Walrand
EECS – U.C. Berkeley
Outline
1. Interpretation2. Probability Space3. Independence4. Bayes5. Random Variable6. Random Variables7. Expectation8. Conditional Expectation9. Notes10. References
1. Interpretation
2. Probability Space2.1. Finite Case
2. Probability Space2.2. General Case
2. Probability Space
3. Independence
Each element has p = 1/4A B
C
4. Bayes’ Rule
B1
B2
A
p1
p2
q1
q2
4. Bayes’ RuleExample:
H0
H1
A = {X > 0.8}
p0
p1
q0
q1
5. Random Variable
x
x0
1
0 1
5. Random Variable
0.5 10.30x
FX(x)
0.210.31
0.650.45
1
5. Random Variable
Slope = afX = 1
a
100
fY = 1/a
5. Random Variable
Other examples:•Bernoulli•Binomial•Geometric•Poisson•Uniform•Exponential•Gaussian
6. Random Variables
6. Random VariablesExample 1
10
Uniform in triangle
X()
Y()
1
0
6. Random VariablesExample 2
xy
g(.)x + dx y + H(x)dx
Scaling by |H(x)|
7. Expectation
0.5 10.30x
FX(x)
0.210.31
0.650.45
1
7. Expectation
Example:
8. Conditional Expectation
8. Conditional Expectation
X
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
10. Reference
Probability and Random Processes