The probability distribution of a random variable $X$ is given below.

$X = x$	$0$	$1$	$2$	$3$
$P(X = x)$	$\frac{1}{10}$	$\frac{2}{10}$	$\frac{3}{10}$	$\frac{4}{10}$

Then the variance of $X$ is

$A$ 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 $X$.

In a communication network, $98 \%$ of messages are transmitted with no error. If a random variable $X$ denotes the number of incorrectly transmitted messages, then the probability that at most one message is transmitted incorrectly out of $500$ messages sent, is

In a hospital, on an average if there are $35$ births in a week, then the probability that there will be less than $3$ births in a day is:

The p.d.f. of a continuous random variable $X$ is given by $f(x) = \frac{x}{8}$ for $0 < x < 4$ and $f(x) = 0$ otherwise. Then $P(X \leq 2)$ is:

$A$ random variable $X$ takes the values $1, 2, 3$ and $4$ such that $2 P(X=1) = 3 P(X=2) = P(X=3) = 5 P(X=4)$. If $\sigma^2$ is the variance and $\mu$ is the mean of $X$, then $\sigma^2 + \mu^2 =$