Note: Differentiating expectations of a function of a random variable with respect to location and scale parameters

Consider a real-valued random variable with a known probability distribution $Pr(z)=\varphi (z)$. From $\varphi (z)$, one can generate a scale/location family of probability densities by scaling and shifting $\varphi (z)$:

$$Pr(x;\mu ,\sigma )=\frac{1}{\sigma }\varphi \left(\frac{x-\mu }{\sigma }\right)$$

The most familiar example of such a family is the univariate Gaussian distribution, when $\varphi (z)=[2\pi {]}^{-1/2}\exp (-\frac{1}{2}{z}^2)$. Now, consider the expectation of a function of $⟨f(x)⟩$ with respect to $Pr\left(x\right)$.

$$⟨f(x)⟩={\int }_{\mathbb{R}}f(x)Pr(x)dx={\int }_{\mathbb{R}}f(x)\frac{1}{\sigma }\varphi \left(\frac{x-\mu }{\sigma }\right)dx$$

What are the derivatives of $⟨f(x)⟩$ with respect to $\mu$ and ${\sigma }^2$? The answers are:

$$\begin{array}{}\text{(1)}& \begin{array}{rl}{\partial }_{\mu }⟨f(x)⟩& =⟨f^{\prime }(x)⟩\\ {\partial }_{{\sigma }^2}⟨f(x)⟩& =\frac{1}{2{\sigma }^2}{⟨(x-\mu )f^{\prime }(x)⟩}_x\end{array}\end{array}$$