Classical Probability is Probability Measure
Jump to navigation
Jump to search
This article, or a section of it, needs explaining, namely:
Link to a page expanding on that statement -- perhaps better to say that it's the definition of the image of Pr?
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this point in more detail, feel free to use the talk page.
This article/section needs proofreading. Please check it for mathematical errors.
If you are fairly certain that there are none, please remove {{proofread}} from the code.
Theorem
The classical probability model is a probability measure.
Proof
We check all the Kolmogorov axioms in turn:
First Axiom
From Empty Set is Subset of All Sets and from the definitions of the event space and sample space:
- $\varnothing \subseteq \Sigma \subseteq \Omega $
From Cardinality of Empty Set and Cardinality of Subset of Finite Set:
- $0 \le \card \Sigma \le \card \Omega$
Dividing all terms by $\card \Omega$:
- $0 \le \dfrac {\card \Sigma} {\card \Omega} \le 1$
The middle term is the asserted definition of $\map \Pr {\, \cdot \,}$.
Link to a page expanding on that statement -- perhaps better to say that it's the definition of the image of Pr?
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this point in more detail, feel free to use the talk page.
If you are able to explain it, then when you have done so you can remove this instance of {{Explain}} from the code.
$\Box$
Second Axiom
- $\map \Pr \Omega = \dfrac {\card \Omega} {\card \Omega} = 1$
$\Box$
Third Axiom
Follows from Cardinality is Additive Function and the corollary to the Inclusion-Exclusion Principle.
$\blacksquare$
If you are fairly certain that there are none, please remove {{proofread}} from the code.
