Probability MT22, Partition theorem
Flashcards
What’s another name for the Law of Total Probability?
The partition theorem.
What does the Law of Total Probability state about a partition of $\Omega$ formed by a family of events $\{B _ 1, B _ 2, \ldots, B _ n\}$ and the probability $\mathbb{P}(A)$ for any $A \in \mathcal{F}$?
\[\mathbb{P}(A) = \sum _ {i \ge 1}\mathbb{P}(A \vert B _ i)\mathbb{P}(B _ i)\]
The Law of Total Probability or partition theorem states that if $\{B _ i, i \in I\}$ is a countable partition of $\Omega$, then
\[\mathbb{P}(A) = \sum _ {i \in I}\mathbb{P}(A \vert B _ i)\mathbb{P}(B _ i)\]
What’s the partition theorem for expectations?
\[\mathbb{E}[X] = \sum _ {i \in I} \mathbb{E}[X \vert B _ i] \mathbb{P}(B _ i)\]
@Prove the partition theorem:
Suppose $B _ 1, B _ 2, \ldots$ is a partition of $\Omega$ by sets from $\mathcal F$, such that $\mathbb P(B _ i) > 0$ for all $i \ge 0$. Then for any $A \in \mathcal F$,
\[\mathbb P(A) = \sum _ {i \ge 1} \mathbb P(A \vert B _ i) \mathbb P(B _ i)\]
Suppose $B _ 1, B _ 2, \ldots$ is a partition of $\Omega$ by sets from $\mathcal F$, such that $\mathbb P(B _ i) > 0$ for all $i \ge 0$. Then for any $A \in \mathcal F$,
@todo (probability, page 10).
@important~