The probability distribution of random variable $X$ is given by:

$X$	$1$	$2$	$3$	$4$	$5$
$P(X)$	$K$	$2K$	$2K$	$3K$	$K$

Let $p=P(1 < X < 4 \mid X < 3)$. If $5p = \lambda K$,then $\lambda$ is equal to .... .

If $X$ is a Poisson variate such that $2 P(X=1)=5 P(X=5)+2 P(X=3)$,then the standard deviation of $X$ is

If $X$ is a Poisson variate such that $\alpha = P(X=1) = P(X=2)$,then $P(X=4)$ is equal to

The range of a random variable $X$ is $\{0, 1, 2\}$. If $P(X = 0) = 3c^3$,$P(X = 1) = 4c - 10c^2$,and $P(X = 2) = 5c - 1$,then find $P(0 < X \le 2)$.

If random variable $X$ is the waiting time in minutes for a bus and the probability density function of $X$ is given by $f(x) = \begin{cases} \frac{1}{5}, & 0 \leq x \leq 5 \\ 0, & \text{otherwise} \end{cases}$,then the probability of the waiting time being not more than $4$ minutes is = . . . . . . .

If the probability function of a random variable $X$ is defined by $P(X=k) = a \left( \frac{k+1}{2^k} \right)$ for $k = 0, 1, 2, 3, 4, 5$,then the probability that $X$ takes a prime value is