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

Result R5308 on D3932: Probability distribution equality relation

Multiplying by the same random real number need not preserve equality in distribution

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} X X \overset{d}{\neq} Y X \end{equation}

Proofs

Proof 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}

The first claim is established in R3923: Standard gaussian random real number is symmetric about zero. For the second claim, we have \begin{equation} \mathbb{P}(X X \geq 0) = \mathbb{P}(X^2 \geq 0) = 1 \end{equation} and \begin{equation} \mathbb{P}(Y X \geq 0) = \mathbb{P}(- X^2 \geq 0) = 0 \end{equation} $\square$