Finding the likelihood ratio hypothesis testing

Question

A single scaled cauchy has density:

$$f_{X\mid θ}(s)=\frac{2}{τ}⋅\frac{\theta}{\theta^2+s^2}$$

Find the likelihood ratio for testing $H_0:θ=1.4$ versus $H_1:θ=2.3$

Suppose that we reject $H_0$ when $f{(X\mid θ_0)}/f(X\mid θ_1)≤1.3$ Solve this inequality for $X$.

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I assume that I plug in the given $\theta$ for each model, but what do I plug in for $s^2$?

Answer 1

$f_{X\,\mid\,\theta}(s)$ is the value at $s$ of the density function of the distribution of the random variable $X.$

$f(X\mid\theta)$ is that same function evaluated at $X,$ so $f(X\mid \theta)$ is itself a random variable.

Since it's the same function, you need to put $X$ where $s$ is in the first line in your question.