What exactly are order statistics? Suppose $X_1,X_2,X_3.....X_n$ are random sample taken from a population.
Then Y(1)<Y(2)<Y(3).....<Y(n) are called order statistics written in increasing order by magnitude where:
Y(1)=minimum($X_1,X_2,X_3.....X_n$)
Y(n)=maximum($X_1,X_2,X_3.....X_n$)
Now the questions are:

*

*How can we find the minimum and maximum of random variable/sample as it's a function, it's not a single number.

*Maybe there is a hint that r.v. are sorted by magnitude. Then what is magnitude of a random variable.
Please explain with real examples.



How can we find the minimum and maximum of random variable/sample as it's a function, it's not a single number.


*

*As a number The maximum of a sample is the highest number of a sample. This is a single number.


*As a function But, when we consider the sample as a random variable that can take different values with different probabilities, then the maximum becomes a random variable that can take different values with different probabilities.
In this case we can describe the maximum with a function that described this probability for a specific value. (and the same for other order statistics)

Example: say you roll two six sided dices and you consider the highest value.

*

*For and particular roll the maximum is the highest number of the roll. For example, if you roll '4' and '1', then the maximum is '4'.


*For the distribution of the possible rolls of the maximum we compute the probabilities for a particular maximum. This will be the probability as function of the value of the maximum.
The figure below shows the maximum as function of the two dice rolls
$$ \begin{array}{c|cccccc}
&\color{red}1 & \color{orange}2  & \color{gold} 3 & \color{green} 4 & \color{blue}5 &\color{purple} 6 \\
\hline 
\color{red}1&\color{red} 1 & \color{orange}2 & \color{gold} 3 & \color{green}4 & \color{blue}5 &\color{purple} 6\\
\color{orange}2&\color{orange}2 &\color{orange} 2 & \color{gold} 3 & \color{green}4 & \color{blue}5 &\color{purple} 6\\
\color{gold} 3&\color{gold} 3 & \color{gold} 3 & \color{gold} 3 &\color{green} 4 & \color{blue}5 & \color{purple}6\\
\color{green}4&\color{green}4 & \color{green}4 & \color{green}4 &\color{green} 4 &\color{blue} 5 & \color{purple}6\\
\color{blue}5&\color{blue}5 &\color{blue} 5 &\color{blue} 5 &\color{blue} 5 &\color{blue} 5 &\color{purple} 6\\
\color{purple}6&\color{purple}6 &\color{purple} 6 & \color{purple}6 &\color{purple} 6 & \color{purple}6 & \color{purple}6\\
\end{array}$$
Then you can see that the probability that the maximum is $1$ is $\frac{1}{36}$, the probability that the maximum is $2$ is $\frac{3}{36}$, the probability that the maximum is $3$ is $\frac{5}{36}$, and so on.
We can describe it as a function
$$ \mathbb{P}(\text{max(2 six sided dice rolls)} = k)  = \frac{2k-1}{36}$$
