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

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Sublevel events do not necessarily coincide for identically distributed random variables

Formulation 0

Let

$a \in \mathbb{R}$ be a D993: Real number.

Then

(1)	$\begin{equation} Z \overset{d}{=} - Z \end{equation}$
(2)	$\begin{equation} \mathbb{P}(Z \leq a) = \mathbb{P}(- Z \leq a) \end{equation}$
(3)	$\begin{equation} \{ Z \leq a \} \neq \{ - Z \leq a \} \end{equation}$

Also known as

Counterexample: identically distributed random variables with distinct sublevel events