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I'm trying to calculate Fisher information (and eventually the Cramer-Rao lower bound) for this particular pdf with Mathematica:

[itex]\text{pte}[t,\Theta ] = \frac{P_{\text{ec}}}{\tau _d-\tau _r}\left[e^{\frac{-(t-\Theta )}{\tau _d}}-e^{\frac{-(t-\Theta )}{\tau _r}}\right]; \text{domain}[\text{pte}] = \{t,-\infty,\infty\}\&\&\{\Theta >0\}[/itex]

So I want to find [itex]I(t|\Theta )[/itex]

This seems relatively straightforward with:

[itex]\text{Integrate}\left[D[\text{Log}[\text{pte}],\Theta ]^2,\{t,-\infty ,\infty \}\right][/itex]

However, Mathematica doesn't want to compute the integral. It just returns the integral itself:

[itex]\int_{-\infty}^{\infty} \frac{\left(\frac{e^{-\frac{t-\Theta }{\tau _d}}}{\tau _d}-\frac{e^{-\frac{t-\Theta }{\tau _r}}}{\tau _r}\right){}^2 \left(\frac{P_{\text{ec}}}{\tau _d-\tau _r}\right)'\left[e^{-\frac{t-\Theta }{\tau _d}}-e^{-\frac{t-\Theta }{\tau _r}}\right]{}^2}{\frac{P_{\text{ec}}}{\tau _d-\tau _r}\left[e^{-\frac{t-\Theta }{\tau _d}}-e^{-\frac{t-\Theta }{\tau _r}}\right]{}^2} \, dt[/itex]

Initially my thought is that there isn't a closed-form solution, but this is something I have seen calculated in journal papers with the same pdf. I'm not an experience Mathematica user. Is there something I have missed in the input (syntax or additional options) or are there any general simplifications that Mathematica would need?

Thanks!

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# Mathematica: Problem Computing Fisher Information of pdf

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