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# Cs221 probability theory

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• A in unobserved, but B is observed
• A in unobserved, but B is observed
• F(x) is monotonically non-decreasing
• PDF is also called probability mass function when applied to discrete random variables
• PDF is also called probability mass function when applied to discrete random variables
• PDF is also called probability mass function when applied to discrete random variables
• PDF is also called probability mass function when applied to discrete random variables
• PDF is also called probability mass function when applied to discrete random variables
• PDF is also called probability mass function when applied to discrete random variables
• PDF is also called probability mass function when applied to discrete random variables
• PDF is also called probability mass function when applied to discrete random variables
• PDF is also called probability mass function when applied to discrete random variables
• ### Cs221 probability theory

1. 1. Probability Theory Review CS221: Introduction to Artificial Intelligence Naran Bayanbat 10/14/2011Slides used material from CME106 course reader and CS229 handouts
2. 2. Topics• Axioms of Probability• Product and chain rules• Bayes Theorem• Random variables• PDFs and CDFs• Expected value and variance
3. 3. Introduction• Sample space - set of all possible outcomes of a random experiment – Dice roll: {1, 2, 3, 4, 5, 6} – Coin toss: {Tails, Heads}• Event space - subsets of elements in a sample space – Dice roll: {1, 2, 3} or {2, 4, 6} – Coin toss: {Tails}
4. 4. Introduction
5. 5. Set operations
6. 6. Conditional Probability A B
7. 7. Conditional Probability Ω A B
8. 8. Conditional Probability
9. 9. Conditional Probability
10. 10. Conditional Probability
11. 11. Conditional ProbabilityP(A, B) 0.005P(B) 0.02P(A|B) 0.25
12. 12. Bayes Theorem
13. 13. Bayes Theorem PosteriorProbability Likelihood Prior Normalizing Probability Constant
14. 14. Bayes Theorem
15. 15. Random Variables Do ya feel lucky, punk?
16. 16. Cumulative Distribution Functions
17. 17. Probability Density Functions
18. 18. Probability Density Functions
19. 19. Probability Density Functions
20. 20. Probability Density Functions f(X) X
21. 21. Probability Density Functions f(X) X
22. 22. Probability Density Functions f(x) x F(x) 1 x
23. 23. Probability Density Functions f(x) x F(x) 1 x
24. 24. Expectation
25. 25. Expectation
26. 26. Variance
27. 27. Gaussian Distributions
28. 28. Questions?