Probability¶
Sample space
Sample space \(\Omega\) defines all the outcomes of a random experiment.
Event space Set of events F denotes all the possible outcomes of an experiment, thus it is a subset of \(\Omega\)
Probability measure
Is a function \(P: F \rightarrow R\) which satisfies the following properties:
\(P(A) \ge 0, \forall A \in F\)
If \(A_1, A_2, \cdots\) are disjoint then \(P(\cup_i A_i) = \sum_i P(A_i)\)
\(P(\Omega) = 1\)
There three are the Axioms of probability from these three we can deriive others:
\(0 \le P(a) \le 1\)
\(P(A \cap B) \le \min(P(A), P(B))\)
Union bound \(P(A \cup B) \le P(A) + P(B)\)
\(P(\Omega - A) = 1 - P(A)\)
In general this function takes an event from the sample space and asigns it to the real line.
Joint probability¶
The join probability of event A and B as: $\( P(A,B) = P(A \cap B) \)$
Marginalization:¶
We sum over all the possible states of one event: $\(P(A) = \sum_b P(A,B=b)\)$
This is also known as the law of total probability.
Conditional probability¶
Let B an event with non-zero probability. The conditional probability of any event A given B is defined as
In other words it is the probability measure of the event A after observing the occurrence of event B.
Union of mutliple events¶
Given event A and B the probabiility that A or B will happen is:
or
If they are mutualy exclusive.
In case we have more than 2 events we follow the inclusion exclusion rule