ThmDex – An index of mathematical definitions, results, and conjectures.
Conditional probability given independent random variable
Formulation 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $X, Y : \Omega \to \Xi$ are each a D202: Random variable on $P$
(ii) $X, Y$ is an D2713: Independent random collection on $P$
(iii) $\{ X \in E \} \in \mathcal{F}$ is an D1716: Event in $P$
Then \begin{equation} \mathbb{P}(X \in E \mid Y) = \mathbb{P}(X \in E) \end{equation}
Proofs
Proof 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $X, Y : \Omega \to \Xi$ are each a D202: Random variable on $P$
(ii) $X, Y$ is an D2713: Independent random collection on $P$
(iii) $\{ X \in E \} \in \mathcal{F}$ is an D1716: Event in $P$
This result is a particular case of R3640: Conditional probability given independent sigma-algebra. $\square$