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

Result R5322 on D3932: Probability distribution equality relation

Adding 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, Z : = - X \end{equation}

Then

(1)	\begin{equation} X \overset{d}{=} Y \end{equation}
(2)	\begin{equation} X + Z \overset{d}{\neq} Y + Z \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, Z : = - X \end{equation}

The first claim is established in R3923: Standard gaussian random real number is symmetric about zero. As for the second claim, we have \begin{equation} X + Z = X - X = 0 \end{equation} surely, as well as \begin{equation} Y + Z = - X - X = - 2 X \end{equation} and clearly $0 \overset{d}{\neq} - 2 X$. $\square$