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

2. Outline Interpretation Probability Space Independence Bayes
Random Variable
Random Variables
Expectation
Conditional Expectation
Notes
References

3. 1. Interpretation

4. 2. Probability Space
2.1. Finite Case

5. 2. Probability Space
2.2. General Case

6. 2. Probability Space

7. 3. Independence
Each element has p = 1/4 A B C

8. 4. Bayes’ Rule B1 B2 A p1 p2 q1 q2

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

10. 5. Random Variable x 1

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

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

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

14. 6. Random Variables

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

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

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

18. 7. Expectation
Example:

19. 8. Conditional Expectation

20. 8. Conditional Expectation

21. 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

22. 10. Reference
Probability and Random Processes