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Content of the lecture

Probability and Statistical Inference (AM Statistics)

Part I: Probability

Random Experiments and Probability

  • Random Experiments

    • Sample Space
    • Events and their Linkage

  • Probabilities

    • Axiomatic Definition of Probability
    • Laplace Experiments
    • Basic Rules of Probability and Combinatorics

  • Conditional Probability

    • Conditional Probability
    • Total Probability and Bayes`Theorem
    • Independence of Events

Random Variables and Distributions

  • Random Variables

    • Cumulative Distribution Function
    • Quantile Function
    • Discrete Random Variables
    • Continuous Random Variables
    • Affine-Linear Transformation of Random Variables

  • Distribution Parameter

    • Expected Value
    • Variance
    • Tchebychev Inequality
    • Skewness

  • Special Discrete Distributions

    • Binomial Distribution
    • Poisson Distribution
    • Geometric Distribution
    • Hypergeometric Distribution
    • Approximation of Discrete Distributions

  • Special Continuous Distributions

    • Rectangular Distribution
    • Exponential Distribution
    • Normal Distribution

Multivariate Distributions and Limit Theorems

  • Joint Distribution of Random Variables

    • Joint and Conditional Distribution of Random Variables
    • Covariance and Coefficient of Correlation
    • Sums of Random Variables

  • Limit Theorems

    • Weak Law of Large Numbers
    • Central Limit Theorem

Part II: Statistical Inference

Samples and Sample Functions

  • Random Sampling
  • Statistics (Functions of Random Samples)
  • Statisticas for Normally Distributed Samples
  • t-Distribution

Method of Estimation for Parameters

  • Estimation of Parameters

    • Point Estimation
    • Unbiasedness and Consistency
    • Maximum-Likelihood-Estimator
    • Estimation of Expected Values
    • Estimation of Probabilities
    • Estimation of Variances and Standard Deviations

  • Interval Estimation of Expected Values and Probabilities

    • Confidence Intervals of Expected Values
    • Confidence Intervals of Probabilities
    • Sample Size

Testing Procedure

  • Estimation of Parameters

    • Testing Problem, Hypotheses
    • Sampling Errors (Type 1 & 2)
    • p-Value

  • Testing of Expected Values

    • One Sample Tests for Expected Values and Probabilities
    • Two Sample Tests for Expected Values and Probabilities

  • Tests for Probabilites

    • Tests for one Probability or Fractile
    • Comparison of two Probabilites or Fractiles

Linear Regression

  • Simple Linear Regression

    • Simple Linear Regression Model
    • Point Estimation of Coefficients
    • Interval Estimation of the Parameters
    • Tests for the Parameters

  • Multiple Linear Regression

    • Interpretation of the Parameters
    • Tests for the Parameter