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I am greatly confused about the Leibniz integral rule for probability densities. If $f_X$ is a valid univariate probability distribution parametrized by $\boldsymbol \theta=\{\theta_1,...,\theta_k\}$, then we should have $$\int_\mathbb{R} f_X(x|\boldsymbol \theta) dx = 1$$ since this is a fundamental property of probability densities. Generally speaking, the value of this integral should not depend on $\boldsymbol \theta$ (assuming the density is defined for all possibles values $\boldsymbol \theta$ can take). So, it seems reasonable that $$\frac{d}{d \theta} \int_\mathbb{R} f_X(x|\boldsymbol \theta) dx=0$$ for all $\theta\in\boldsymbol\theta$. However, if I use the Leibniz integral rule, I get the strange result that $$\frac{d}{d \theta} \int_\mathbb{R} f_X(x|\boldsymbol \theta) dx= \int_\mathbb{R} \frac{\partial}{\partial \theta} f_X(x|\boldsymbol \theta) dx\neq0$$ since the derivative $\frac{\partial}{\partial\theta}f_X(x|\boldsymbol\theta)$ is not $0$ if $\theta$ is a proper parameter of $f_X$.

Where did I make a mistake? I think that all necessary assumptions for the Leibniz rule should be fulfilled, given that we are in a measurable probability space.

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    $\begingroup$ Did you try a concrete example? $\endgroup$ Sep 2, 2022 at 10:21
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    $\begingroup$ You are not guaranteed that $\frac{\partial}{\partial\theta}f_X(x\mid \boldsymbol\theta)$ exists for all $x$: consider a uniform distribution on $[0,\theta]$ for $\theta>0$ and what happens to the partial derivative at $x=\theta$. If it does exist for all $x$, the partial derivative will sometimes be positive and sometimes negative - how do you know it does not integrate to $0$? $\endgroup$
    – Henry
    Sep 2, 2022 at 10:47
  • $\begingroup$ @Henry Oh, that's it. I forgot to consider that the derivative can be negative. Cheers. $\endgroup$
    – mto_19
    Sep 2, 2022 at 10:50

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From comment as requested

You are not guaranteed that $\frac{\partial}{\partial\theta}f_X(x\mid \boldsymbol\theta)$ exists for all $x$: consider a uniform distribution on $[0,θ]$ for $θ>0$ and what happens to the partial derivative at $x=θ$.

If it does exist for all $x$, the partial derivative will sometimes be positive and sometimes negative - how do you know it does not integrate to $0$?

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