Summary 20592 - Statistics Notes (for DSBA Bocconi)

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Statistics
Probability Recap
Definition of Probability
Probability measures and axioms
Independent Events
Conditional Probabilities
Bayes’ Theorem
Random Variables
Point mass distribution
Discrete uniform distribution
Bernoulli distribution
Binomial distribution
Negative Binomial
Geometric RV
Poisson distribution
Continuous Uniform distribution
Normal (Gaussian) distribution
Exponential distribution
Gamma distribution
Beta distribution
t distribution
Cauchy distribution
The χ2 distribution
Weibull distribution
Bivariate distributions
Marginal distributions



Statistics 1

, Independent RV’s
Conditional distributions
Multinomial distribution
Multivariate Normal
Transformations of RV’s
Expectation
Properties of Expectations
Variance
Sample moments
Covariance
Conditional Moments
Probability Inequalities
Markov’s Inequality
Chebyshev’s Inequality
Cauchy-Schwartz Inequality
Jensen Inequality
Convergence of RV’s
Almost Sure Convergence
Convergence in Probability
Convergence in Distribution
Convergence in quadratic mean
Convergence in rth (or Lr) mean
Relationships of convergences
The Law of Large Numbers
The Central Limit Theorem
The Delta Method
Computational Methods 1
Integration techniques
Numerical Integration technique
The Newton-Cotes quadrature
Monte Carlo techniques
The Inverse Transform method
The Acceptance-Rejection method
Importance Sampling
Classical Statistical Inference
Classical Statistical inference procedure
Point Estimation
Confidence sets
Hypothesis Testing
Empirical Distribution Function
Statistical Functionals
Bootstrap
Parametric Inference




Statistics 2

, Properties of the MLE
Asymptotic normality of MLE
Hypothesis Testing and p-values
Neyman and Pearson strategy
Most Powerful test
Wald Test
Hypothesis testing with two samples
Multiple Testing
Computational Methods 2
Newton’s Method
Generalized Linear model
Fisher scoring algorithm
Expectation and Maximization (EM) algorithm
Inference for stochastic processes: Markov Chains
White Noise
Summaries
Stationarity
Markov Chains
Ergodicity
Inference for Markov chains
Elements of Bayesian Learning
Credible intervals
Predictive distribution
Computational Methods 3
Metropolis Hasting algorithm
The algorithm
Detailed balance condition
Independent MH algorithm
Random Walk chains
Gibbs Sampler



Probability Recap
Definition of Probability
Probability is a mathematical language for quantifying uncertainty.
The sample space Ωis the set of possible outcomes of an experiment. Points ω in
Ω are called sample outcomes, realizations, or elements. Subsets of Ωare called
elements.
If we toss a coin forever, then the sample space is the infinite set




Statistics 3

, Ω = {ω = (ω1 , ω2 , ω3 , … ) : ωi ∈ {H, T }}
​ ​ ​ ​




The elements are all the possible numbers of tosses with all the possible
combinations of H and T.
e.g.
ω1 contains 3 tosses, with all the possible combinations of H and T for 3 tosses.
​




If every element of A is also contained in B, we write A ⊂ B.




A1 , A2 , …are disjoint or are mutually exclusive if Ai ∩ Aj = ∅whenever i =
​ ​



 j . ​




Given an event A, define the indicator function of A by:


IA (ω) = I(ω ∈ A) = {
1 if ω∈A
0 if ω∈
​ ​




/A

A sequence of sets A1 , A2 , …is monotone increasing if A1
​ ​ ​ ⊂ A2 ⊂ …and we
​




define:

∞
lim An = ⋃ Ai
​ ​ ​ ​




n→∞
i=1

Same for monotone decreasing.


Probability measures and axioms
A function Pthat assigns a real number P(A)to each event A is a probability
distribution or a probability measure if it satisfies the following three axioms.

1. P(A) ≥ 0 ∀A


Statistics 4