Combinatorics. Elements of the theory of probability: discrete and continuous random variables, probability distributions and densities. Populations, statistical samples, mean, median, variance. Multivariate samples, covariance matrix. PCA and clustering. Point estimators and confidence intervals, hypothesis testing.
Lecture notes and exercises available on the teachers' web page.
P. Baldi – Calcolo delle Probabilita' e Statistica.
S.M.Ross – Calcolo delle probabilita' e Statistica.
Basic knowledge for the statistical treatment of data.
Fundamentals of probability.
To set up a statistical analysis of data.
Skills acquired (at the end of the course):
Construction of models related to simple problems of calculation of the probabilities' and their solution.
To compute confidence intervals; run a hypothesis test; use ACP and Cluster analysis.
Courses to be used as requirements (required and/or recommended)
Courses required: none
Courses recommended: none
Contact hours for: Lectures (hours): 48
Type of Assessment
Oral exam. The exam generally consists in one exercise and two theoretical questions on Probability and Statistics.
Combinatorics, Frequentist, subjective, axiomatic definition off probability '. Conditional probability, Bayes Theorem. Independence. Random variables. Distribution functions and probability density function. Expected value of a R.V., variance. Functions of random variables.
Most common discrete and continuous probability distributions.
Normal distribution; normalization of a random variable. Joint distributions, marginal density function. Chebychev inequality. Almost sure convergence and convergence in probability. Law of large numbers.
Individuals, population , character . Types of character. Statistical sample . Mode , modal values ??, median, arithmetic mean and sample variance .Bivariate samples . Covariance , correlation coefficient, linear regression line . Reminders on carriers Analysis of Principal Component ( PCA). Distance between individuals. Distance between clusters . Hierarchical clustering .Statistical sample . Statistics. Sample mean and sample variance . Some properties of Gaussian distributions . Central Limit Theorem .Chi square and Student's t distributions and their links with the Gaussian samples. Confidence intervals.Confidence intervals ( bilateral and unilateral ) for the mean of Gaussian samples with known variance. Choice of sample size .Confidence intervals (bilateral and unilateral) for the mean of the Gaussian sample of which is not known variance. Confidence intervals ( bilateral and unilateral ) for the variance of the Gaussian samples. Hypothesis testing, bilateral and unilateral , for the mean of Gaussian samples with known variance .Hypothesis testing, bilateral and unilateral , for the mean of the Gaussian sample variance unknown.Hypothesis testing for the equality, of the means of Gaussian samples with known variance .Hypothesis testing for the equality of the means of Gaussian samples with unknown but equal variances. Hypothesis testing (approximate) for the equality of the means of Gaussian samples with variances unknown and different . One-sided hypothesis testing for the variance of the Gaussian samples. Hypothesis testing for discrete density . Normality test.