Meaning of estimation? When we have an estimate of random variable $X$ in terms of random variable $Y$ we can write it as $\hat{X}=f(Y)$ where $\hat X$ is the estimate and $f$ is a function.
Then suppose we observe a sample $y\in Y$. Then do we just plug in the sample $y$ to $f$ i.e., $f(y)$ to find the sample $x$ of  $X$ which gave us $y$? 
If so please explain to me what the reasoning behind plugging in a sample in place of a random variable? My confusion is the random variable $Y$ is a function but $y$ is just a value.
Thanks a lot.
 A: When an experiment is conducted (or a superior being conducts the experiment
and allows you to observe the results), all the random variables take on
values. For example, $X$ might have value $\beta$ while $Y$ has value $\alpha$
on this particular trial of the experiment.
 In your problem, the value of $Y$ is observable while the value of $X$ is hidden
from you, and you are being asked to estimate the value of $X$ on this particular trial; that is, you have to estimate $\beta$, the value of $X$ on
this particular trial.
$\hat{X} = f(Y)$ means that if $Y$ has value $\gamma$ on this particular
trial, then the random variable $\hat{X}$ has value $f(\gamma)$ on this
trial of the experiment. So, since you observe that $Y$ has value 
$\alpha$ on a particular trial of the experiment, you know that
$\hat{X}$ has value $f(\alpha)$ on this experiment, and you are using
this number as an estimate of $\beta$, the unknown (and unobservable)
value $\beta$ that $X$ has on this same trial.  How good this estimate 
$f(\alpha)$ is of the unknown value $\beta$ is a different question.
