Probability and Statistics I
Objectives
General characterization
Code
10975
Credits
6.0
Responsible teacher
Carlos Manuel Agra Coelho
Hours
Weekly - 4
Total - 70
Teaching language
Português
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 obtain the final approval for the course, either by Tests or Exam, the students will need to obtain a rate of attendance of at least 2/3 to Labs.
2. Evaluation
- The recommended form of evaluation consists in 2 Tests:
- 1st Test - weight: 50% (0-20 scale)
- 2nd Test - weight: 50% (0-20 scale) - with a minimum grade of 7
- Students will be approved in case they obtain from tests a final grade of 9.5 or more (on a 0-20 scale).
- Students who obtained a final grade from tests less than 9.5 (on a 0-20 scale), may have access to a final Exam.
- 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: