Carefully defining all italicised terms, show that, if a sufficiently general method of inference respects both the Weak Sufficiency Principle and the Conditionality Principle, then it respects the Likelihood Principle.

The position $X_{t}$ of a particle at time $t>0$ has the Normal distribution $\mathcal{N}(0, \phi t)$, where $\phi$ is the value of an unknown parameter $\Phi$; and the time, $T_{x}$, at which the particle first reaches position $x \neq 0$ has probability density function

$$p_{x}(t)=\frac{|x|}{\sqrt{2 \pi \phi t^{3}}} \exp \left(-\frac{x^{2}}{2 \phi t}\right) \quad(t>0) .$$

Experimenter $E_{1}$ observes $X_{\tau}$, and experimenter $E_{2}$ observes $T_{\xi}$, where $\tau>0, \xi \neq 0$ are fixed in advance. It turns out that $T_{\xi}=\tau$. What does the Likelihood Principle say about the inferences about $\Phi$ to be made by the two experimenters?

$E_{1}$ bases his inference about $\Phi$ on the distribution and observed value of $X_{\tau}^{2} / \tau$, while $E_{2}$ bases her inference on the distribution and observed value of $\xi^{2} / T_{\xi}$. Show that these choices respect the Likelihood Principle.