Probability and Statistics I

General characterization

Prerequisites

The students should be provided with basic knowledge about calculus (mathematical analysis: geometric and arithmetic progressions, sumations, series, derivation and integration)

Bibliography

Coelho, C. A. (2008). Tópicos em Probabilidades e Estatística, Vol. I, Vol. II (Cap.^s 6,7).

Mood, A. M., Graybill, F. A. e Boes, D. C. (1974). Introduction to the Theory of Statistics, 3ª ed., J. Wiley & Sons, New York.

Montgomery, D. C. e Runger, G. C. (1998). Applied Statistics and Probability for Engineers, 2ª ed., J. Wiley & Sons, New York.

Ross, S. M. (1999). Introduction to Probability and Statistics for Engineers and Scientists. J Wiley & Sons, New York.

Murteira, B. J. F. (1990). Probabilidades e Estatística, Vol I, 2ª ed., McGraw-Hill Portugal, Lisboa.

Rohatgi, V. K. (1976). An Introduction to Probability Theory and Mathematical Statistics. J. Wiley & Sons, New York.

Teaching method

• 2 weekly Theoretical Classes (in a total of 3 hours per week), where the concepts are introduced and analysed and the main results are derived and proven. Illustrative examples are also shown.
• 1 weekly Lab of 2 hours where exercises and problems pertaining the concepts and results shown in the Thoeretical classes are solved.

Evaluation method

 1. Pre-Requisites

In order to be able to have access to the course evaluation, both to midterms and tests and also to the Exam, students on a first enrolment need to have the presence in at least 80% of both Labs and Classes, being this percentage reduced to 2/3 of both Labs and Classes for the other students (once obtained, this presence score, in case it will be necessary, wil remain valid for the following year).

 2. Evaluation

• The recommended form of evaluation consists in 2 Tests:
• 1st Test - weight: 40% -  April 8 (allowed and necessary the use of a simple calculator, which cannot be a graphic one)
• 2nd Test - weight: 60% -  June 12 (no calculator allowed)
• The student who has an average grade of at least 9.5 (on a 0-20 scale) will be approved in the course.
• Students who obtained a final grade from tests less than 9.5 (on a 0-20 scale), may have access to a final Exam, in case they have attended at least 2/3 of Labs and 2/3 of Classes.
• Also the students who had a grade equal or greater than 9.5 from tests may have access to the Final Exam in order to improve their grade.
• Students with a final grade of more than 17 (on a 0-20 scale) have to go through an oral examination, or their final grade will be equal to 17.

Subject matter

1 – Combinatorics (short review) 

2 – Elementary Probability Theory

• Random experiment and Outcome space
• Event and Event Space
• The concept of Probability. Probabilities Properties
• Conditional Probability and independence of events
  • Some useful and iteresting results on Conditional Probabilities
  • Conditional Independence and (Marginal) Independence
  • Odds and Odds ratio
  • Illustrative example of the reason of the definition of the Independence of 3 or more events
  • Examples of application (of the notion of conditional probability, Bayes formula and Bayes and Total Probability Theorems)
  • The Borel-Cantelli Lemmas

3 – Random variables and Probability Distributions

• Definition of random variable. Examples
• Probability Distribution functions. Properties. Quantiles
• The Survival and Risk functions – two alternative ways of representing the distribution of a r.v.
  • Development and study of a Risk function
  • Survival and Risk functions for discrete r.v.''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''s
• Expected value. Properties. Moments. Some important inequalities involving moments
• Moment generating functions and characteristic functions
• The distribution of Y = g(X)

4 – Joint and conditional distributions of random variables

• Joint distribution of two or more random variables
• Joint and marginal moments
• The joint moment generating function
• Conditional distributions and independence
  • Conditional moments
  • The conditional expected value
  • Some additional notes on the conditional expected value
  • Independence of r.v.s
  • Consequences of independence
  • Other conditional distributions
  • Truncated distributions as conditional distributions
• Joint distributions of r.v.s of different types
• The distribution of (Y1, Y2) = g(X1,X2)
• The distributions of Sum, Difference, Product and Ratio of two r.v.s
• Mixtures

5 – Discrete random variables

• The Uniform distribution
• The Geometric distribution
• The Negative Binomial distribution
• The Bernoulli distribution
• The Binomial distribution
• The Hipergeometric distribution
• The Poisson distribution

6 – Continuous random variables

• The Exponential distribution
• The Normal distribution
• The chi-square distribution
• The T distribution
• The F distribution
• The Gamma distribution

7 – Brief reference to multivariate distributions

• The Multinomial distribution
• The Multivariate Normal distribution

Programs

Programs where the course is taught: