– Set Theory: Basic concepts and set operations. Relations between sets.
– Probability: Axiomatic introduction to the concept of probability. Random experiments – Sample space. The laws of probability. Counting methods – Combinatorics. Conditional probability and independent events.
– Discrete Random Variables: Definition, Expected value (mean), Moments, Variance, Standard deviation. Cumulative distribution function. Moment-generating function.
– Continuous Random Variables: Definition, Probability density function, Mean, Variance and moments, Moment-generating function, Functions of a continuous random variable.
– Basic Distributions of Discrete and Continuous Random Variables: Bernoulli distribution, Binomial distribution, Geometric distribution, Negative binomial distribution, Hypergeometric distribution, Poisson distribution, Uniform distribution, Exponential distribution, Gamma and χ² distributions, Normal distribution, Lognormal distribution.
– Multivariate Random Variables: Two-dimensional random variables, Conditional distributions, Covariance and correlation coefficient. Independent random variables. Functions of random variables. Probability density function of a function of two random variables. Joint probability density function of two functions of two random variables.
– Sample: Sample mean and variance. Sample variance from a normal distribution. Student’s t distribution. F distribution. Central Limit Theorem.
– Statistics: Introduction. Basic Definitions: Population, Sample, Variable, Data. Classification and presentation of statistical data. Classification parameters: Frequency, Relative frequency and Relative frequency %, Cumulative frequency, Relative cumulative frequency and Relative cumulative frequency %. Tables. Diagrams.
– Measures of Location: Mean, Median, Mode, Percentiles, Quartiles. Measures of Dispersion: Range, Interquartile range, Variance and standard deviation, Coefficient of variation. Z-scores.
– Moments: General moments, Moments about the origin, Central moments. Skewness and measures of skewness. Kurtosis and measures of kurtosis. Boxplot.
– Estimation: Point estimation. Criteria for good estimators. Methods for calculating point estimates: Method of moments, Maximum likelihood method.
– Confidence Intervals: Confidence intervals for the mean, for the difference of means, for the variance, for the difference of proportions.
– Hypothesis Testing: p-value of a test. Hypothesis tests for means, for the difference of means, for the variance, for the difference of proportions.