How to show an ancillary statistic is a first order ancillary statistic?

Question

A statistic $A(X)$ is first order ancillary if $\mathbb{E}_{\theta}[A(X)]$ does not depend on $\theta$ where $X \sim P_{\theta}$. Show when distribution of $A(X)$ is independent of $\theta$, $\mathbb{E}_{\theta}[A(X)]$ is independent of $\theta$?

My try:

I think we need to show $\mathbb{E}_{\theta}[A(X)] = \int_{-\infty}^{+\infty}A(x)p_{\theta}(x)dx$ is independent of $\theta$ but how?

Answer 1

By the definition of an ancillary statistic, the function $p_\theta$ is the same for all $\theta$, therefore so is the integral.