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

Result R4598 on D5480: Even real function
Subresult of R4597: Centred gaussian real density function is an even function

Standard gaussian real density function is an even function

Formulation 0

Let $f : \mathbb{R} \to \mathbb{R}$ be a D2864: Real gaussian density function such that

(i)	\begin{equation} f(t) = \frac{1}{\sqrt{2 \pi}} e^{- \frac{1}{2} t^2} \end{equation}

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

Then \begin{equation} f(-x) = f(x) \end{equation}

Proofs

Proof 0

Let $f : \mathbb{R} \to \mathbb{R}$ be a D2864: Real gaussian density function such that

(i)	\begin{equation} f(t) = \frac{1}{\sqrt{2 \pi}} e^{- \frac{1}{2} t^2} \end{equation}

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

This result is a particular case of R4597: Centred gaussian real density function is an even function. $\square$