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

$X = x$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$
$P(X = x)$	$0$	$k$	$2k$	$2k$	$3k$	$k^2$	$2k^2$	$7k^2 + k$

Then, $P(0 < X < 4)$ is equal to:

The cumulative distribution function (c.d.f.) $F(x)$ of a discrete random variable $X$ is given by the following table:

$X$	$-3$	$-1$	$0$	$1$	$3$	$5$	$7$	$9$
$F(X=x)$	$0.1$	$0.3$	$0.5$	$0.65$	$0.75$	$0.85$	$0.90$	$1$

Then,find the value of $\frac{P[X=-3]}{P[X < 0]}$.

$A$ service station manager sells gas at an average of ₹ $100$ per hour on a rainy day,₹ $150$ per hour on a dubious day,₹ $250$ per hour on a fair day,and ₹ $300$ per hour on a clear sky day. If weather bureau statistics show the probabilities of weather as follows,then his mathematical expectation is:

Weather	Clear	Fair	Dubious	Rainy
Probability	$0.50$	$0.30$	$0.15$	$0.05$

$A$ 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 $X$.

An observer counts $240$ vehicles per hour at a specific location on a highway. Assuming that the arrival of vehicles at the location follows a Poisson distribution, the probability that more than two vehicles arrive over a $30 \text{ sec}$ time interval is

$A$ random variate $X$ takes the values $0, 1, 2, 3$ and its mean is $1.3$. If $P(X=3) = 2 P(X=1)$ and $P(X=2) = 0.3$,then $P(X=0)$ is equal to: