Probability Function

up:: 023 MOC Measure Theory

A probability function is a function defined from the Sigma-algebra of a Measurable Space $(X,\mathcal{F})$ to the real numbers, $P:\mathcal{F}\to \mathbb{R}$ which satisfies

• $\forall A\in \mathcal{F}:P(A)\ge 0$;
• $P(X)=1$
• For a countable set of mutually disjoint measurable sets $\{A_n{\}}_{n\in \mathbb{N}}$, we have

$$P\left(\bigcup _{n\in \mathbb{N}}{A}_n\right)=\sum _{n\in \mathbb{N}}P(A_n)$$

Properties

• $\forall A\in \mathcal{F}:P(A^c)=1-P(A)$¹
  • $⟹P(\varnothing )=0$
• $\forall A\subseteq B\in \mathcal{F}:P(A)\le P(B)$²
  • $⟹\forall A\in \mathcal{F}:0\le P(A)\le P(X)=1$
• $\forall A,B\in \mathcal{F}:P(A\cup B)=P(A)+P(B)-P(A\cap B)$³
• Continuity Theorem of Probabilities over (non-)decreasing sets of events

---

References

• Hogg, Robert V., Joseph W. McKean, and Allen T. Craig. Introduction to mathematical statistics. Pearson Education India, 2013.

Footnotes

1. Since $C=A\cup A^c$, and since $P(A)+P(A^c)=1$. ↩

2. Note that $B=A\cup (A^c\cap B)$ of disjoint sets; thus, $P(B)=P(A)+P(\dots )\ge P(A)$. ↩

3. Note that $A\cup B=A\cup (B\setminus A)=A\cup (B\cap A^c)$ and $B=(B\setminus A)\cup A=(B\cap A^c)\cup (B\cap A)$ is a union of disjoint parts, from which $P(B\cap A^c)=P(B)-P(A\cap B)$. ↩