This series of tutorials is an introduction to the basic statistics used in machine learning algorithms.
Definitions: let (\(\Omega\) , F, P) be a measure space with P(\(\Omega\)) = 1. This means that (\(\Omega\) , F, P) is a probability space where \(\Omega\) = sample space, F = event space and P = Probability measure.
In Kolmogorov's probability theory the probability P of some event E must satisft 3 axioms.
Probability of Empty Set:
P(\(\emptyset\)) = 0
A \(\subseteq\) B then P(A) \(\subseteq\) P(B)
0 \(\leq\) P(E) \(\leq\) 1
Probability of Union of Events
P(A \(\cup\) B) = P(A) + P(B) - P(A \(\cap\) B)
It can also be shown as a consequence of axiom 3 the following:
P(A \(\cup\) B) = P(A) + P(B \ (A \(\cap\) B)). In other words the event A or B can be described as the event A occuring or only B occuring.
Principal of Inclusion Exclusion
P(\(A^c\)) = P(\(\Omega\)\A) = 1 - P(A)
This is just a very basic introduction to statistics and will be the foundation on which further theorems will build upon. I highly recommend the reader to proof the consequences outlined above. If you have any question please leave a comment below.