ThmDex – An index of mathematical definitions, results, and conjectures.

Identically distributed random variables need not be almost surely equal

Formulation 0

Let $X \in \text{Gaussian}(0, 1)$ be a D211: Standard gaussian random real number such that

(i)	\begin{equation} Y : = - X \end{equation}

Then

(1)	\begin{equation} X \overset{d}{=} Y \end{equation}
(2)	\begin{equation} \mathbb{P}(X \neq Y) = 1 \end{equation}

Formulation 1

Let $X \in \text{Gaussian}(0, 1)$ be a D211: Standard gaussian random real number such that

(i)	\begin{equation} Y : = - X \end{equation}

Then

(1)	\begin{equation} X \overset{d}{=} Y \end{equation}
(2)	\begin{equation} X \overset{a.s.}{\neq} Y \end{equation}