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

Result R4597 on D5480: Even real function

Centred 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 \sigma^2}} e^{- \frac{1}{2} \big( \frac{t}{\sigma} \big)^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 \sigma^2}} e^{- \frac{1}{2} \big( \frac{t}{\sigma} \big)^2} \end{equation}

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

Since we have \begin{equation} \left( \frac{-t}{\sigma} \right)^2 = \frac{(-t)^2}{\sigma^2} = \frac{(-1)^2 t^2}{\sigma^2} = \frac{t^2}{\sigma^2} \end{equation} the claim follows. $\square$