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

(A) There are $15$ students in the class. Each student has the same chance to be chosen. Therefore,the probability of each student being selected is $\frac{1}{15}$. The frequency distribution of ages is as follows:

$X$	$14$	$15$	$16$	$17$	$18$	$19$	$20$	$21$
$f$	$2$	$1$	$2$	$3$	$1$	$2$	$3$	$1$

The probability distribution $P(X)$ is:

$X$	$14$	$15$	$16$	$17$	$18$	$19$	$20$	$21$
$P(X)$	$\frac{2}{15}$	$\frac{1}{15}$	$\frac{2}{15}$	$\frac{3}{15}$	$\frac{1}{15}$	$\frac{2}{15}$	$\frac{3}{15}$	$\frac{1}{15}$

Mean $E(X) = \sum X_i P(X_i) = \frac{1}{15}(14 \times 2 + 15 \times 1 + 16 \times 2 + 17 \times 3 + 18 \times 1 + 19 \times 2 + 20 \times 3 + 21 \times 1) = \frac{263}{15} \approx 17.53$.
Mean of squares $E(X^2) = \sum X_i^2 P(X_i) = \frac{1}{15}(196 \times 2 + 225 \times 1 + 256 \times 2 + 289 \times 3 + 324 \times 1 + 361 \times 2 + 400 \times 3 + 441 \times 1) = \frac{4683}{15} = 312.2$.
Variance $(X) = E(X^2) - [E(X)]^2 = 312.2 - (17.5333)^2 = 312.2 - 307.4177 = 4.7823 \approx 4.78$.
Standard deviation $= \sqrt{\text{Variance}} = \sqrt{4.7823} \approx 2.19$.