Let X be a single observation from the density f(x;$\theta$)= $(2\theta x + 1 - \theta)$ I$_{[0,1]}$(x), where $-1<=\theta<=1$. Find the likelihood ratio test with the significance level $\alpha$ of H$_o$: $\theta=0$ versus H$_1$: $\theta=1$
My approach: $\lambda$ = $\frac {f(x;0)}{f(x;1)}$ = $\frac {1}{2x}$ <k we reject H$_o$
meaning x>= $\frac {1}{2k}$ we reject H$_o$
$\alpha$ = P(x>= $\frac {1}{2k}$) = $ \int\limits_\frac {1}{2k}^1 (2\theta x + 1 - \theta) \ dx $ =$1-\frac{\theta}{4k^2}+\frac{\theta}{2k}-\frac{1}{2k}$
Based on this i want to know when do we reject H$_o$
For convenience let me work with $C=1/(2k)$. You've shown that the rejection region is $X>C$. Good. The key to determining $C$ is to focus on the null distribution, with $\theta=0$. Note that we will want $C\in(0,1)$, otherwise we'll get $0$ or $1$ for the probability below. Since $f(x;0)=1$ for $x\in[0,1]$, we want $$\alpha\stackrel{?}{=}P_{\theta=0}(X>C)=\int_C^1 1dx=1-C,$$ thus the level $\alpha$ LR test rejects $H_0$ when $X>1-\alpha$.