# Probabilities and Statistics

Computer Engineering, Publication in the Diário da República - Despacho n.º 8644/2020 - 08/09/2020

5 ECTS; 2º Ano, 1º Semestre, 28,0 PL + 28,0 TP

Prerequisites
Not applicable.

Objectives
It is intended that students achieve in the course unit of Probabilities and Statistics the learning results:
a) recover and consolidate knowledge of Probabilities;
b) acquire knowledge about random variables and some theoretical distributions of probability (discrete and continuous);
c) acquire knowledge and develop mathematical skills in the estimation (punctual and interval) and decision, as well as in the study of the relationship between two variables (correlation and linear regression);
d) use the acquired knowledge and skills developed to design and implement solutions to various problems applied under conditions of uncertainty.

Program
1 Probability
1.1 Notion of probability
1.2 Probability and frequency: law of large numbers
1.3 Random experiences and events
1.4 Definition of probability of an Event
1.5 Axioms of probability
1.6 Union of events and additive rules
1.7 Conditional probability and events independence
1.8 Intersection of events and multiplicative rules
1.9 The Total probability theorem
1.10 Bayes' theorem

2 Random variables
2.1 Discrete and continuous random variables
2.2 Discrete probability distributions
2.3 Continuous probability distributions
2.4 Functions of random variables
2.5 Expected value and variance of a random variable

3 Some probability distributions
3.1 Discrete distributions: uniform, Bernoulli, binomial, geometric and Poisson distribution
3.2 Continuous distributions: uniform, normal, exponential, gamma distribution
3.3 Relationship between distributions

4 Sampling and sampling distributions
4.1 Population and sample. Sampling Methods
4.2 Most common sample statistics
4.3 Distribution of the sample mean. central limit theorem
4.4 Distribution of sample variance
4.5 Distribution of the sample proportion

5 Parameter estimation
5.1 Estimator and Estimate
5.2 Methods for determining estimators
5.3 Properties of estimators
5.4 Point and interval estimation
5.5 Mean confidence interval (known population standard deviation)
5.6 Student's t distribution
5.7 Mean confidence interval (population standard deviation unknown)
5.8 Chi-square distribution
5.9 Standard deviation and variance confidence interval
5.10 Confidence intervals of proportions

6 Hypothesis tests
6.1 Null hypothesis and alternative hypothesis
6.2 Test statistics
6.3 Critical region
6.4 Bilateral and unilateral tests
6.5 First and second type errors
6.6 Power of a test
6.7 Testing the expected value of a population
6.8 Tests to variances
6.9 Tests to proportions

7 Correlation and Regression
7.1 Scatter diagram
7.2 Simple linear regression model. Minimum squares method
7.3 Analysis of variance: ANOVA
7.4 Determination and correlation coefficients
7.5 Response prediction

Evaluation Methodology
Continuous assessment: two written tests during the semester (marked from 0 to 20) subject to a minimum mark of 6 marks in both. Assessed subject content is distributed throughout the two. A minimum overall mark of 10/20 of tests exempts students from sitting the exam.
Exam-based Assessment (normal assessment period): written test covering all the material taught in the module (graded from 0 to 20).
Remaining periods: written test covering all the material taught in the module ( marked from 0 to 20).

Bibliography
- Gama, S. e Pedrosa, A. (2004). Introdução Computacional à Probabilidade e Estatística. Porto - Portugal: Porto Editora
- Cabral, J. e Guimarães, R. (2007). Estatística. Lisboa - Portugal: McGraw-Hill
- Grilo, L. (2013). Probabilidades e Estatística. Conceitos Teórico-Práticos. Instituto Politécnico de Tomar, Portugal: Instituto Politécnico de Tomar

Method of interaction
The teaching methodology of this curricular unit consists of theoretical-practical classes with oral exposition and examples (assisted with notes), as well as laboratory classes, where several application exercises are solve

Software used in class
Occasionally, Excel and the SPSS statistical package can be used to solve some exercises.