Probability and Statistics cheat sheet

Đây là cheat sheet mình dùng ôn thi môn XSTK tuần rồi. Do là cheat sheet nên nội dung ngắn gọn. Thực ra còn một số phần trong hypothesis testing và limit theorems nhưng mình không đưa vào vì dài quá. Hi vọng tuần này thi xong sẽ có thời gian viết về vài chủ đề trong này.

Nội dung gồm:

1. Random experiments
• Sample space and probability
• Combinatorics
• Conditional probability for events
2. Random variables
• Definition
• CDF, PDF
• Moments
• Multivariate RVs: Joint probability density (mass) function, Cumulative joint distribution function, Marginal density (mass) function, Moments, Conditional probability function, Conditional cumulative distribution, independence.
3. Probability distributions
• Uniform
• Bernoulli
• Binomial
• Poisson
• Exponential
• Gaussian
4. Generating functions and Moment generating functions
5. Sampling
6. Point estimation
• MLE
• MAP
• Lest-square estimation
7. Interval estimation
• $X ~ \mathcal{N}\left(\mu, \sigma^2\right)$, estimate $\mu$ with known $\sigma^2$
• $X ~ \mathcal{N}\left(\mu, \sigma^2\right)$, estimate $\mu$ with unknown $\sigma^2$
• $X ~ \mathcal{N}\left(\mu, \sigma^2\right)$, estimate $\sigma^2$ with both $\mu, \sigma^2$ unknown
• Estimate the difference in mean
8. Hypothesis Testing
9. Limit theorems
• Chebysev’s inequality
• Convergence in law (distribution), probability, mean-square, “almost-sure”
• Weak/Strong Law of Large Number
• Central Limit Theorem