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.
Previous exams:

Spring-2005