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., McGrawHill 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. PreRequisites
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% (020 scale)
 2nd Test  weight: 50% (020 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 020 scale).
 Students who obtained a final grade from tests less than 9.5 (on a 020 scale), may have access to a final Exam.
 Students with a final grade of more than 17 (on a 020 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 BorelCantelli 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 chisquare 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: