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