A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16, 18, 20, 17, 16, 19 years. One student is selected in such a manner that each has the same chance of 5 being chosen and the age $X$ of the selected student is recorded. What is the probability distribution of the random variable $X$ ? Find mean, variance and standard deviation of $X$ .

.: $X$ denote the random variable which represents the ages of 15 students.

$\therefore$ $X$ denote the random variable which represents the ages of 15 students.

$\therefore$ $X$ can assume values 14, 15, 16, 17, 18, 19, 20 and 21 can assume values 14, 15, 16, 17, 18, 19, 20 and 21

Hence, the probability distribution is:

Also, Mean $= \sum X P(X) = \cfrac{{263}}{{15}} = 17.53$
${\sigma _X} = \sqrt {E\left( {{X^2}} \right) - {{(E(X))}^2}} = \sqrt {\cfrac{{4683}}{{15}} - \cfrac{{69169}}{{225}}} = \sqrt {478} = 2.19$