Probability & Statistical Inference Fundamentals
Formula-first lookup tables for rapid recall
1. Probability Foundations
Probability Spaces
| Concept | Formula |
|---|
| Sample space | |
| Event | |
| Probability measure | |
| Normalization | |
| Additivity | |
Conditional Probability & Independence
| Concept | Formula |
|---|
| Conditional probability | |
| Independence | |
| Conditional independence | |
| Bayes' rule | |
Random Variables
| Concept | Formula |
|---|
| Random variable | |
| Discrete PMF | |
| Continuous PDF | with |
| CDF | |
2. Expectation & Moments
Expectation
| Concept | Formula |
|---|
| Expectation (discrete) | |
| Expectation (continuous) | |
| Linearity | |
Variance & Covariance
| Concept | Formula |
|---|
| Variance | |
| Alternate form | |
| Covariance | |
| Correlation | |
Laws
| Law | Formula |
|---|
| Total probability | |
| Total expectation | |
| Total variance | |
3. Common Distributions
Discrete
| Distribution | PMF | Mean | Variance |
|---|
| Bernoulli | | | |
| Binomial | | | |
| Poisson | | | |
Continuous
| Distribution | PDF | Mean | Variance |
|---|
| Uniform | | | |
| Normal | | | |
| Exponential | | | |
4. Inequalities & Concentration
| Inequality | Statement |
|---|
| Markov | |
| Chebyshev | |
| Hoeffding | |
5. Sampling & Limit Theorems
Laws of Large Numbers
| Type | Statement |
|---|
| Weak LLN | |
| Strong LLN | |
Central Limit Theorem
6. Estimation Theory
Point Estimation
| Concept | Formula |
|---|
| Sample mean | |
| Sample variance | |
| Bias | |
| MSE | |
Likelihood & MLE
| Concept | Formula |
|---|
| Likelihood | |
| Log-likelihood | |
| MLE | |
7. Confidence Intervals
Mean Estimation
| Case | Interval |
|---|
| Known | |
| Unknown | |
Variance Estimation
| Interval | Formula |
|---|
| Variance CI | |
8. Hypothesis Testing
General Structure
| Step | Description |
|---|
| Null hypothesis | |
| Alternative | |
| Test statistic | Function of data |
| Significance | |
| Decision | Reject / fail to reject |
Common Tests
| Test | Statistic | Reference |
|---|
| Z-test | | |
| One-sample t | | |
| Two-sample t | | Welch |
| Chi-square | | |
Errors & Power
| Concept | Definition |
|---|
| Type I error | Reject true |
| Type II error | Fail to reject false |
| Power | |
| p-value | |