# Maths for Experimental Sciences

**Descriptive statistics**

Introduction: problems of descriptive and inferential statistics and their applications in the experimental sciences.

Elements of descriptive statistics: discrete variables and continuous variables, population; character; sample; absolute frequency; relative frequency; cumulative frequency; variable; statistics; dot plot; bar graph, pie chart.

Main statistics: mode, median, quartiles, quantiles, arithmetic mean, deviation.

Statistical averages: definition of Cauchy; definition of Chisini; the arithmetic mean; geometric mean; harmonic mean;

weighted arithmetic mean; their properties (proofs and applications).

Variability indexes: the range of the data; deviance; variance and standard deviation; coefficient of variation; their properties (proofs and applications). .

Form of a distribution: the concept of symmetry; asymmetry; the standardized variable; Pearson index of asymmetry; Fisher asymmetry index; Kurtosis and Pearson kurtosis index.

Analysis of dependence: two-variable data table; conditional distribution and independence; scatterplots; Chi-Square for contengency table; Cramer's V index; linear regression and least squares technique; association; regression coefficient and its interpretation; variance of regression and its decomposition; coefficient of determination; polynomial regression; linearization methods.

Analysis of interdependence: the concept of interdependence; measure of concordance; discordance; the correlation coefficient of Bravais -Pearson and its interpretation.

Elements of partial correlation and multiple regression: plane and hyperplane of regression; partial regression coefficient; coefficient of determination; partial correlation coefficient .

Examples and applications in bio-pharmaceutical sciences

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**Elements of probability**

Combinatorial analysis: the basic principle of counting; permutations (theorem and application); combinations (theorem and application); binomial coefficient, dispositions (theorem and application),

The algebra of Events: standard event, implication between events, equal events; complement of an event; the union event; the intersection event; mutually exclusive events; sample space; conditional events.

The different definitions of probability: classical, frequentist, axiomatic; some simple propositions on the probability of events. Sample spaces having equally likely outcomes; card games; the problem of birthdays.

Introduction to conditional probability: the multiplication rule; independent events; Bayes' Formula (proof and application).

The random variables: random variable (r.v.); distribution function; probability mass function; density function; the Bernoulli r.v.; the Binomial r.v.; the Normal r.v.; the Standard Normal r.v.; the Student's t r.v.; moments of a r.v..

Limit theorems: the Central Limit Theorem; the weak law of large

numbers; the strong law of large numbers.

**Elements of hypothesis testing**

Hypothesis Testing: basic concept of random samples: research hypothesis; statistical hypothesis, null hypothesis; alternative hypothesis (uni and bi-directional); statistical significance; statistical distribution of the test statistic, critical values, rejection regions and acceptance region, error probabilities and the power function; the Z test for a sample mean; the T test for a sample mean; degrees of freedom; confidence intervals for the mean; sample size determination; p-values.

Examples and applications in bio-pharmaceutical sciences.