<span role="img" aria-label="theoretical-statistics-emoji" style="margin-right:10px">📈 Theoretical Statistics

1.2

Conditional probability

1.3

Independence

1.4

Bayes' theorem

1.5

Combinatorics

Unit 2 – Random Variables and Probability Distributions

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2.1

Discrete random variables

12 min read

2.2

Continuous random variables

2.3

Probability mass functions

2.4

Probability density functions

2.5

Cumulative distribution functions

2.6

Common probability distributions

Unit 3 – Expectation and moments

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3.1

Expected value

3.2

Variance and standard deviation

3.3

Covariance and correlation

3.4

Moment generating functions

3.5

Higher-order moments

Unit 4 – Multivariate distributions

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4.1

Joint probability distributions

4.2

Marginal distributions

5 min read

4.3

Conditional distributions

4.4

Multivariate normal distribution

4.5

Transformations of random vectors

Unit 5 – Limit Theorems and Convergence in Statistics

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5.1

Law of large numbers

5.2

Central limit theorem

5.3

Types of convergence

5.4

Asymptotic theory

5.5

Delta method

Unit 6 – Statistical inference

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6.1

Population and sample

6.2

Sampling distributions

6.3

Point estimation

6.4

Interval estimation

6.5

Maximum likelihood estimation

Unit 7 – Estimation theory

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7.1

Properties of estimators

7.2

Sufficiency

7.3

Completeness

7.4

Rao-Blackwell theorem

7.5

Cramer-Rao lower bound

Unit 8 – Hypothesis testing

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8.1

Null and alternative hypotheses

8.2

Type I and Type II errors

8.3

Power of a test

8.4

Likelihood ratio tests

8.5

Multiple testing

Unit 9 – Bayesian statistics

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9.1

Bayesian inference

9.2

Prior and posterior distributions

9.3

Conjugate priors

9.4

Bayesian estimation

9.5

Bayesian hypothesis testing

Unit 10 – Stochastic processes

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10.1

Markov chains

10.2

Poisson processes

5 min read

10.3

Brownian motion

10.4

Martingales

10.5

Time series analysis

Unit 11 – Sampling theory

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11.1

Simple random sampling

11.2

Stratified sampling

5 min read

11.3

Cluster sampling

11.4

Systematic sampling

11.5

Sample size determination

Unit 12 – Decision theory

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12.1

Decision rules

12.2

Loss functions

12.3

Risk and Bayes risk

12.4

Minimax decision rules

12.5

Admissibility and completeness