One addition to the title: For $(\Omega,\cal{F},P)$, $\Omega=\mathbb{R}$.
Thanks in advance.
I hope there are some others than only Gaussian (with same variance!).
One addition to the title: For $(\Omega,\cal{F},P)$, $\Omega=\mathbb{R}$.
Thanks in advance.
I hope there are some others than only Gaussian (with same variance!).
There are many examples, but for example take two normal distributions with the same variance $\sigma^2$ and means $\mu_0 \lt \mu_1$ where $L = \exp\left(x\frac{(\mu_1-\mu_0) }{\sigma^2}\right)\exp\left(\frac{\mu_0^2-\mu_1^2}{2\sigma^2}\right).$
If the conditional densities $f_1$ and $f_0$ are exponential densities with parameters $\lambda_1$ and $\lambda_0$ respectively where $\lambda_0 > \lambda_1$, then $L(x) = \frac{f_1(x)}{f_0(x)} = \frac{\lambda_1\exp(-\lambda_1x)}{\lambda_0\exp(-\lambda_0x} = \frac{\lambda_1}{\lambda_0}\exp((\lambda_0-\lambda_1)x)$ is an increasing function of $x$ on $[0,\infty)$ that increases from $\frac{\lambda_1}{\lambda_0} < 1$ to $\infty$. Does that work for you?