Download Probability

Basic Probability
Jean Walrand
EECS – U.C. Berkeley
Outline
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Interpretation
Probability Space
Independence
Bayes
Random Variable
Random Variables
Expectation
Conditional Expectation
Notes
References
1. Interpretation
2. Probability Space
2.1. Finite Case
2. Probability Space
2.2. General Case
2. Probability Space
3. Independence
C
A
Each element has p = 1/4
B
4. Bayes’ Rule
B1
q1
p1
q2
p2
B2
A
4. Bayes’ Rule
Example:
H0
q0
p0
q1
p1
H1
A = {X > 0.8}
5. Random Variable
1
x
0
0
x
1
5. Random Variable
FX(x)
1
0.31
0.65
0.45
0.21
x
0
0.3
0.5
1
5. Random Variable
fY = 1/a
a
Slope = a
0
fX = 1
0
1
5. Random Variable
Other examples:
•Bernoulli
•Binomial
•Geometric
•Poisson
•Uniform
•Exponential
•Gaussian
6. Random Variables
6. Random Variables
Example 1
1
Uniform in triangle
w
Y(w)
0
0
X(w)
1
6. Random Variables
Example 2
x + dx
y + H(x)dx
g(.)
y
x
Scaling by |H(x)|
7. Expectation
FX(x)
1
0.31
0.65
0.45
0.21
x
0
0.3
0.5
1
7. Expectation
Example:
8. Conditional Expectation
8. Conditional Expectation
X
9. Notes
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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(w) 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