============================
 Probability and Statistics
============================

What is the probability of throwing a 6 with a fair die? Almost
everyone immediately will answer: $1/6$ -th. But what do we mean by
that? Two obvious lines of reasoning to arrive at that answer are:

- If we throw a die there are six possible outcomes and because it is
  a fair die every outcome is equally probable, the probabilities of
  each possible outcome should add up to one, and so the probability
  of throwing a 6 is equal to $1/6$ -th.

- Let's repeat the experiment a large number of times, say $N$. After
  throwing the die $N$ times we estimate the probability as:

  .. math::
     \text{Probability(throwing 6)} = \frac{\text{#\{throwing 6\}}}{N}

  This estimate will approach the true probability if we let
  $N\rightarrow\infty$.


This line of reasoning is part of what is called the **frequentist**
approach to probability. We will follow this route quite often in this
course.

But now consider the question: what is the probability that it will
rain tomorrow? Neither of the two lines of reasoning above can help us
here? Tomorrow it will either rain or not but these possible outcomes
are clearly not necessarily equally probable. And if we repeat the
experiment $N$ times (say we take $N=10\times365$, then we have data
over 10 years) and use the frequentist approach we undoubtly end up
with the probability that it will rain on any randomly given day in
the Netherlands. Certainly not the probability that it will rain
*tomorrow*. Assigning a probability of 70% for rain tomorrow
quantifies our **belief** that it will rain tomorrow. A belief that is
based on data and models.

Fortunately the interpretation of probability (being a frequentist
view or a belief based view) is of lesser importance. The classical
mathematical rules for probability and statistics that we will look at
in this course are largely independent of the interpretation. Loosely
speaking we might say that **probability is the mathematical language
of choice when dealing with uncertainty.**

Probabilty theory deals with random experiments and random
processes. Experiments and processes that are not deterministic, it is
simply not possible to know beforehand exactly what will be the result
of such an experiment or process. In probability theory we assume that
for each of the possible outcomes of a random experiment the
corresponding probabilities are well defined.



.. figure:: /figures/kaasboor.jpg
   :align: right
   :figwidth: 30%

   The Dutch term "steekproef" for a statistical sample, comes from
   the way Dutch cheese is tested on the traditional cheesemarket in
   Alkmaar. A cylindrical sample out of a cheese is tested to assess
   the quality of several cheeses.

With **statistics** we enter the realm of everyday life. We may assume
that there exists a function that assigns a probability to each
possible outcome of a random experiment, but alas all we have are
observations (a sample or 'steekproef' in Dutch) of the random
experiment. Statistics then deals (among others) with the question
what can be known about the underlying random experiment using only
the observations in the sample? Because the sample consists of random
numbers the conclusions from statistics about the random experiment
are in a sense random as well. We can never be completely sure about
our (numerical) conclusions. A lot a statistics deals with this
important question: can we quantize the probability that we arrive
at the right (or wrong) conclusions?





.. toctree::

   probSpaceAxioms
   probConditional
   probTrees
   probIndependence
   probRVs
   probDistributions
   probJointDistributions
   rvCalculations
   probStatistics
   probEstimation
   probRandomVectors
   probExercises