From experiments to
probability distributions of random variables
Experiments, outcomes, events
Probability: {events} → [0, 1]
(combinatorial methods, Bayes theorem...)
Random variables: {outcomes} → ℝ
Probability distribution of a random variable
Why are
probability distributions so important
To infer, on a statistical base, the processes
underlying an experiment
To make predictions and
implement statistical tests
Our main
goal!
An approach to investigate and characterize
probability distributions
Expected values
Moments of a distribution
A circular argument: to evaluate expected values
(and moments) one has to know the distribution!
So... to estimate distribution parameters,
2
like  and  , use statistics (functions of data)
that are estimators, like x and s2
Very powerful tools ...
Chebyshev inequality
Sample mean, x
Sample variance, s2
Central limit theorem
… and applications
Value and error of a measurement via x and s