Conditional Probability Defines Probability Space
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Theorem
Let $\struct {\Omega, \Sigma, \Pr}$ be a measure space.
Let $B \in \Sigma$ such that $\map \Pr B > 0$.
Let $Q: \Sigma \to \R$ be the real-valued function defined as:
- $\map Q A = \condprob A B$
where:
- $\condprob A B = \dfrac {\map \Pr {A \cap B} }{\map \Pr B}$
is the conditional probability of $A$ given $B$.
Then $\struct {\Omega, \Sigma, Q}$ is a probability space.
Proof
It is to be shown that $Q$ is a probability measure on $\left({\Omega, \Sigma}\right)$.
As $\Pr$ is a measure, we have that:
- $\forall A \in \Omega: \map Q A \ge 0$
Also, we have that:
\(\ds \map Q \Omega\) | \(=\) | \(\ds \condprob \Omega B\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\map \Pr {\Omega \cap B} } {\Pr \left({B}\right)}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\map \Pr B} {\map \Pr B}\) | Intersection with Universe | |||||||||||
\(\ds \) | \(=\) | \(\ds 1\) | as $\map \Pr B > 0$ |
Now, suppose that $A_1, A_2, \ldots$ are disjoint events in $\Sigma$.
Then:
\(\ds \map Q {\bigcup_{i \mathop = 1}^\infty A_i}\) | \(=\) | \(\ds \frac 1 {\map \Pr B} \map \Pr {\paren {\bigcup_{i \mathop = 1}^\infty A_i} \cap B}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {\map \Pr B} \map \Pr {\bigcup_{i \mathop = 1}^\infty \paren {A_i \cap B} }\) | Intersection Distributes over Union | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i \mathop = 1}^\infty \frac {\map \Pr {A_i \cap B} } {\map \Pr B}\) | as $\Pr$ is a measure | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i \mathop = 1}^\infty \map Q {A_i}\) |
$\blacksquare$
Sources
- 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $\S 1.6$: Theorem $1 \text {A}$