Statistics is a field of Mathematics primarily concerned with Probability, it's definition and interpretation as well as figuring out a whole lot of things about how probabilities behave.

In it's most Technical form Probability is concerned with mathematical objects that adhere to Kolmogorovs Axioms. In the Following the A and B are the Mathematical objects adhering to the Axioms and P(X) is the Probability. U is the space of all possible states of the System.

Kolmogorovs Axioms

$$ P(A) \ge 0$$

$$ P(U) = 1$$

$$P(A \cup B) = P(A) + P(B)\quad\forall A, B;\ A\cap B = \emptyset$$

Note that These axioms do not make any statement about how to interpret the objects that adhere to these axioms. There are two geneally used Interpretations

Interpretations of Probability

  • Frquentist interpretation the fequentist interpretation Sees probability as the limit of a specific event A occurring in a total of N events where N tends towards inifinitely many events. So in short the frequentist interpretation of probability is: $$ P(A) = \lim_{ N \to \infty } \frac{ N(A)}{N} $$ It is important to add that P(A) now also depends on how the space U is sampled. So it is important to specify how one came to the sample A of which the probability is to be / has been calculated.

  • Bayesian interpretation The core of the bayesian interpretation is the idea of the degree of beleif That is in other words the degree of certainty. It makes the probability 'personal', saying that probability is something that human minds have constructed and that does not exist in 'reality'. In this context it is often used in conjuntion with and in relationship to gambling, as it is a more or less exact fit for that set of circumstances. Probability in the bayesian sense is nearly allways connected to some cost of a decision and the wish to maximise profit whilst avoiding loss by making certain decisions. How cost/reward is calculated depends on the individual making the decision. The same numbers can therefore result in different actions, because the costs/benefits are weighted differently.

The Probability density function

The probability density function or pdf is the central quantity of interest in physics. The pdf describes the probability distribution and all other quantities can be calculated from it. Most to all work (at least in high energy physics comes down to somehow predicting, calculating and measuring various probability density functions.