# Probability and Statistics

## Course Description:

The meaning of probability. Axioms of probability. Conditional probability. Independent events. Random variables. Density and cumulative distribution functions. Parameters of distributions. Generating and characteristic functions. Special discrete and continuous distributions. Functions of random variables. Central limit theorem. Random sample and sampling distributions. Estimation of parameters. Point estimation. Interval estimation. Hypothesis testing. Goodness of fit tests. Contingency tables. Simple and multiple linear regressions.

### Prerequisite Knowledge

It is recommended for the students to have the basic background in Mathematical Analysis I and II.

### Course Units

# Title Description Hours
1 Introduction Random experiments, Historical review, Set theory. 2
2 Introduction to Probability Sample space, Events, Definitions of probability, Axioms of probability, Conditional probability, Independence, Elements of combinatorics, Exercises. 10
3 Univariate Random Variables Discrete and continuous random variables, Distribution function, Probability mass function, Probability density function, Exersices. 4
4 Moments of Random Variables Expectation, Variance, Standard deviation, Moments, Exercises. 4
5 Specific Discrete Distributions Bernoulli, Binomial, Geometric, Negative Binomial, Hypergeometric, Poisson, Exercises. 4
6 Specific Continuous Distributions Uniform, Normal, Exponential, Gamma, Weibull, X2, Exercises. 4
7 Functions of Random Variables Discrete case, Continuous case, Distribution of the sum of random variables, Distribution of the maximum and minimum, Exercises. 4
8 Central Limit Theorem Approximate distribution of independent and identically distributed random variables, Approximate distribution of the sample mean of independent and identically distributed random variables, Approximation of Binomial by the Normal distribution, Exercises. 4
9 Introduction to Statistics Introduction to the problem. Random sample and sampling distributions. Elements of descriptive statistics. 2
10 Point Estimation Unbiasedness, Method of moments, Method of maximum likelihood, Exercises. 8
11 Confidence Intervals Introductory notions and definitions, Interpretation of a confidence interval, Applications based on the normal distribution, Approximate confidence intervals, Connection to hypothesis testing, Exercises. 6

### Learning Objectives

The module Probability - Statistics delivers an introduction in the modeling and analysis of stochastic systems. Its aim is to familiarize the students with the notions of random variables, distributions and parameters of them, as well as to develop skills in stochastic quantitative calculations. Additionally, methods of estimating unknown quantities are introduced, using Statistical techniques, with the help of a random sample.

### Teaching Methods

 Teaching methods Lectures in class. Implementations using examples, exercises and case studies. Blackboard

### Student Assessment

• Final written exam: 70%
• Mid-term exam: 30%

### Textbooks - Bibliography

1. Kokolakis, G. & Spiliotis, I (2010). Probability Theory and Statistics with Applications. Symeon. Athens (In Greek).
2. Vonta, I. & Karagrigoriou, A. (2017). Applied Statistical Analysis and Elements of Probability. Paraskinio. Athens (In Greek).
3. Fouskakis, D. (2013). Data Analysis using R. Tsotras. Athens (In Greek).
4. Kokolakis, G. & Fouskakis, D. (2009). Statistical Theory and Applications. Symeon. Athens (In Greek).
5. Bertsekas, D. & Tsitsiklis, G. (2013). Introduction to Probability and Elements of Statistics. Tziolas. Athens (In Greek).
6. Roussas, G. (2011). Introduction to Probability Theory. Ziti. Athens.
7. Hoel, P.G., Port, S.C. & Stone, C.J. (2009). Introduction To Probability Theory. University of Crete, Heraklion (In Greek).
8. Ross, S. (2011). Basic elements of Probability Theory. Klidarithmos. Athens (In Greek).
9. Spiegel, M.R. (Translation Persidis, S.) (1977). Probability and Statistics. Espi. Αthens (In Greek).
10. Class notes and Exercises in the following url : http://www.math.ntua.gr/~fouskakis/civil-eng.html.

## Lecture Time - Place:

• Tuesday, 08:45 – 10:30,
Rooms:
• Ζ. Κτ. 1 Πολ., Αιθ. 1
• Thursday, 12:45 – 14:30,
Rooms:
• Αμφ. 1/2

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