Introduction
We may measure a random variable indirectly. However, the indirect measurement is affected by the noise following a certain distribution. Thus, the measurement is represented by a random variable that follows a certain distribution. For example, we want to measure $\theta$, the temperate of the stomach, and it can be measured by measuring the mouth’s temperature $Y$. $$Y = \theta + W$$ $W$ is the noise that follows the normal distribution $N\sim(0,\sigma)$. If $Y$ is very informative regarding $\theta$, $Y$ with different $\theta$ should be distinguishable. If $\sigma$ is large, the distribution of $Y$ is plattered, and the $Y$ with different $\theta$ will overlap largely, which is less distinguishable. The less indicative $Y$ contains less information about $\theta$. How informative the random variable $Y$ is regarding $\theta$ can be measured using Fisher information.
$$ I_{Y}(\theta)=E\left[\left(\frac{\partial}{\partial \theta} \ln f(Y ; \theta)\right)^{2}\right] $$