The Standard Normal Distribution Has Finite Fisher Information, Score Minus the Identity, and Fisher Information One

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, in dimension $d=1$, with the real line identified with $\mathbb{R}^{1}$ as in One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative, whose reading of differentiability, of $f'$ and of continuity, and whose identification of $\lVert x\rVert^{2}$ with $x^{2}$, are in force. Let $\lambda$ be Lebesgue measure $\lambda_{1}$ on $\mathcal{B}(\mathbb{R})$, which is the Lebesgue measure of the real line by Lebesgue Measure on $\mathbb{R}^n$. Let $N\in\mathcal{P}(\mathbb{R})$ be the standard normal distribution, with the function $g(x)=\exp(-x^{2}/2)$ and the positive real number $c=\int_{\mathbb{R}}g\,d\lambda$ as there, and define $\varphi:\mathbb{R}\to\mathbb{R}$ by $\varphi(x)=c^{-1}g(x)$. Class $C^{1}$ on $\mathbb{R}=\mathbb{R}^{1}$ and the partial derivative $\partial_{1}$ are those of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (ii), $\mathbb{R}^{1}$ being open by claim 1 of Euclidean Space Is Open in Itself. Let $M_{2}(N)$ be the second moment of $N$, let $\mathrm{id}$ be the identity map of $\mathbb{R}$, and, $N$ having finite second moment by claim 2, let finite Fisher information, the set $\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R})$, the score $\xi_{N}$ and the Fisher information $\mathcal{I}(N)$ be as defined there.

1. (Density)(i) The function $\varphi$ is of class $C^{1}$ on $\mathbb{R}$, hence continuous and Borel, it is differentiable at every point of $\mathbb{R}$, and for every $x\in\mathbb{R}$

$$\varphi(x)>0,\qquad \varphi'(x)=\partial_{1}\varphi(x)=-x\,\varphi(x);$$

and $N$ is the measure with density $\varphi$ with respect to $\lambda$, that is, $N(B)=\int_{\mathbb{R}}\mathbf{1}_{B}\,\varphi\,d\lambda$ for every $B\in\mathcal{B}(\mathbb{R})$.

2. (Second moment)(ii) $M_{2}(N)=\int_{\mathbb{R}}x^{2}\,N(dx)=1$; in particular $N\in\mathcal{P}_{2}(\mathbb{R})$.

3. (Score)(iii) $N\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R})$, the score of $N$ is $\xi_{N}=-\mathrm{id}$, and $\mathcal{I}(N)=1$.