ThmDex – An index of mathematical definitions, results, and conjectures.
P2689
Since $X_0 : = Z = : X_1$, then clearly $X_0$ and $X_1$ agree in distribution. Furthermore, result R3923: Standard gaussian random real number is symmetric about zero implies that \begin{equation} X_2 : = - Z \overset{d}{=} Z = : X_1 \end{equation} This establishes the first claim. Next \begin{equation} \begin{split} \mathbb{P}(X_0 \leq a, X_1 \leq a) & = \mathbb{P}(Z \leq a, Z \leq a) \\ & = \mathbb{P}(\{ Z \leq a \} \cap \{ Z \leq a \}) \\ & = \mathbb{P}(\{ Z \leq a \}) \\ & = \mathbb{P}(Z \leq a) \\ & = \mathbb{P}(Z \in (-\infty, a]) \\ & \neq \mathbb{P}(Z \in [-a, a]) \\ & = \mathbb{P}(- a \leq Z \leq a) \\ & = \mathbb{P}(Z \leq a, Z \geq -a) \\ & = \mathbb{P}(Z \leq a, - Z \leq a) \\ & = \mathbb{P}(X_1 \leq a, X_2 \leq a) \end{split} \end{equation} $\square$