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1 |
Matrix algebra - Dimensions and common matrix types
- Matrix addition, scalar multiplication, and matrix multiplication
- Transpose, determinants, and inverses
- Rank, eigenvalues, eigenvectors, and diagonalisation
- Systems of equations and matrix differentiation
- Chain rule, OLS, gradients, Jacobians, Hessians, and the variance of the OLS estimator
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2 |
Probability and discrete random variables - Sets and probability axioms
- Addition rule, joint and conditional probability, and independence
- Bayes’ theorem and random variables
- Joint and marginal distributions for discrete variables
- Covariance, correlation, conditional expectation, and expectation rules
- Bernoulli, binomial, and Poisson distributions
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3 |
Continuous random variables and distributions - CDFs and probability density functions
- Uniform distribution
- Expectation and variance
- Joint and marginal distributions, covariance, and independence
- Conditional density and conditional expectation
- Normal, chi-squared, t, and exponential distributions
- Degrees of freedom
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4 |
Population, sampling, and limit theory - Population and sample; population and sampling distributions
- Law of large numbers
- Markov and Chebyshev inequalities
- Central limit theorem
- Big-O and little-o notation
- Moment-generating and characteristic functions
- Sampling distribution of the sample variance
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5 |
Estimation and hypothesis testing - Point and interval estimation
- Confidence intervals with known and unknown variance
- Confidence intervals when the population distribution is unknown
- Construction and interpretation of hypothesis tests
- One-sided and two-sided tests
- Testing differences in means
- Type I and Type II errors
- p-values
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