Probability

[Intro to Probability & Basic Concepts]
Kolmogorov's Axioms, DeMorgan's Laws

Expected Value & Variance & Covariance

Bayes' Theorem

Independent Event

Inequalities - Proofs and Examples 
- Markov Inequality, Chebyshev's Inequality, Cauchy-Schwarz' Inequality 

Convergence of Random Variables

- Convergence in Probability / Distribution, Almost Sure Convergence, the Central Limit Teorem
 

How to Estimate Parameter?

 - Intro to statistical inference, MoM and MLE 

 

Sufficient Statistics, Factorization Theorem, Exponential Family
 

Hypotesis Testing
- Neyman-Pearson Lemma, Uniform Most Powerful Test, Likelihood Ratio Test
 
Variance/ Bias Tradeoff  
- The Hill Estimator, Kernel Density Estimation, Non-parametric Regression 

 

[Discrete Random Variable]

$\triangleright$  Bernoulli Distribution

 - Mean, Variance, MLE, Sufficient Statistics, Exponential Family  

$\triangleright$ Binomial Distribution 
 - Mean, Variance, MLE, Hypothesis Testing 

$\triangleright$ Geometric Distribution 
- Mean, Variance, MoM, MLE, Confidence Interval 

$\triangleright$ Poisson Distribution 
- Mean, Variance, MLE, Confidence Interval 


[Continuous Random Variable]
$\triangleright$ Uniform Distribution 

$\triangleright$ Exponential Distribution 


[Parametric Distribution] 
$\triangleright$ The Chi-squared, t, and F distributions
- Definition