Probability and Statistical Methods

The topics of this exam are related to the courses Statistical Methods, ESMA 6305, and Probability Theory, ESMA 6600.

Topics in Probability:

  • Sample spaces and events.
  • Axiomatic definition of probability.
  • Conditional probability, sequence of dependent and independent events.
  • Expected value and variance of a discrete random variable.
  • Distributions of discrete random variables: Binomial, Geometric, Poisson.
  • Expected value and variance of a continuos random variable.
  • Distributions of continuous random variables: Uniform, Gamma, Exponential and Normal.
  • Distribution of functions of random variables.
  • Moment generating function.
  • Bi-dimensional random variables (both variables dicrete or both variables continuous), joint distributions, marginal distributions, conditional distributions, conditional expectations, conditional variance, covariance and correlation.
  • Limit Theorems.
  • Markov Chains, Transition Matrix and related properties.

Topics in Statistical Methods:

  • Review of classical random variables distributions: discrete and continuous.
  • Sampling distributions related to normal distributions: t-student, Chi-square, F.
  • Principles related to inference (confidence, precision, type I error, type II error, power of a statistical test, etc.)
  • Inference based on one-sample for the mean and the variance: Confidence intervals and hypothesis testing.
  • Inference based on two-samples for the means and the variances: Confidence intervals and hypothesis testing.
  • Inference based on two dependent samples.
  • Analysis of categorical data: (goodness of fit, contingency tables, independence and homogeneity test).
  • Correlation.
  • Linear Regression.
  • Basic Principles of Experimental designs: (compute the ANOVA table, the parameters of the model, verify the model assumptions, perform multiple comparisons). Some Experimental designs may include: One-way analysis of variance, two-way analysis of variance, Factorial design.
  • Multiple linear regression and correlation.
  • Basic nonparametric tests.

Bibliography (Probability Part):

  • Modern Mathematical Statistics, E.J. Dudewicz and S.N. Mishra, (1988) John Wiley, NY.
  • Probability, W. Feller, (1978) John Wiley, NY.
  • Probability Theory, R.G. Laha, (1976) John Wiley, NY.
  • An Introduction to Probability Models, S. Ross, (2002) 8th Ed. Academic Press, Boston.
  • A first course in probability, S. Ross, (2001) 6th Ed. Prentice Hall.

Bibliography (Statistical Methods Part):

  • Statistical Methods of Analysis, Chin Long Chian, (2003), World Scientific Publishing.
  • Statistics for Research, S. Dowdy, S. Waerden and S. Chiko, 3rd Ed. John Wiley, NY.
  • Modern Applied Statistics with S-plus, B. Venables and B.D.R. Ripley, Springer-Verlag.
  • Statistics for Applied Solving Problems and Decision Making, R. Larsen, M. Marx and B. Cooli, (1997) Duxbury Press, Boston.
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