Jean Walrand EECS – U.C. Berkeley Basic Probability Jean Walrand EECS – U.C. Berkeley

Outline 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 Each element has p = 1/4 A B C

4. Bayes’ Rule B1 B2 A p1 p2 q1 q2

4. Bayes’ Rule Example: H0 H1 A = {X > 0.8} p0 p1 q0 q1

5. Random Variable x 1

5. Random Variable 0.5 1 0.3 x FX(x) 0.21 0.31 0.65 0.45

5. Random Variable Slope = a fX = 1 a 1 fY = 1/a

5. Random Variable Other examples: Bernoulli Binomial Geometric Poisson Uniform Exponential Gaussian

6. Random Variables

6. Random Variables Example 1 w 1 Uniform in triangle X(w) Y(w)

6. Random Variables Example 2 g(.) y + H(x)dx x + dx y x Scaling by |H(x)|

7. Expectation 0.5 1 0.3 x FX(x) 0.21 0.31 0.65 0.45

7. Expectation Example:

8. Conditional Expectation

8. Conditional Expectation

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(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