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1 What is Probability? 3
1.1 Sigma Algebra, Borel Sets, etc.. . . . . . . . . . . . 3
1.2 Experiment.. . . . . 3
1.2.1 Examples.. . 3
1.3 Event.. . . . . . . . 3
1.3.1 Example.. . . 3
1.4 Sample Space.. . . . 3
1.4.1 Example.. . . 3
1.5 Complementary Event. . . . . . . . . . . . . . . . . 3
2 Discrete Probability 3
2.1 Probability of Two Events. . . . . . . . . . . . . . . 3
2.1.1 Example.. . . 4
2.1.2 Union Probability (OR). . . . . . . . . . . . 4
2.1.3 Joint Probability (AND). . . . . . . . . . . . 4
2.1.4 Mutually Exclusive. . . . . . . . . . . . . . . 4
2.2 Conditional (Posterior) Probability. . . . . . . . . . 5
2.3 Bayes' Theorem.. . . 5
2.4 Total Probability.. . 5
3 Distributions 5
3.1 Binomial Distribution.5
3.2 Multivariable Gaussian. . . . . . . . . . . . . . . . . 6
3.3 Summary.. . . . . . 6
3.4 Variance.. . . . . . . 6
3.5 Covariance.. . . . . . 6
3.5.1 Correlation Coecient. . . . . . . . . . . . . 6
3.5.2 Sample Covariance. . . . . . . . . . . . . . . 7
4 Continuous Probability 8
4.1 Conditional Distribution. . . . . . . . . . . . . . . . 8
4.2 Conditional Distribution 2. . . . . . . . . . . . . . . 8
4.3 Independence.. . . . 8
4.4 Bayes Theorem.. . . 8
4.4.1 Total Probability. . . . . . . . . . . . . . . . 8
4.5 Cumulative Distribution Function. . . . . . . . . . . 8
4.6 Variance.. . . . . . . 9
5 Random Variables 10
5.1 Expected Value.. . . 10
5.2 Conditional Mean.. . 10
5.3 Conditional Expected Value. . . . . . . . . . . . . . 10
5.4 Uncorrelated.. . . . . 10
5.5 Orthogonal.. . . . . 10
6 Moments 10
6.1 First Moment - Mean.10
6.2 Second Moment - Variance. . . . . . . . . . . . . . . 11
6.3 Third Moment - Skew. . . . . . . . . . . . . . . . . 11
6.4 Fourth Moment - Kurtosis. . . . . . . . . . . . . . . 11
6.5 Central Moments.. . 11
7 Functions of a Random Variable 11
7.1 Example.. . . . . . . 12
8 2D Distributions 12
8.1 Joint Distributions.. 12
8.2 Marginal Distribution.12
8.3 2D CDF.. . . . . . . 12