For a Poisson variate $X$,if $P(X=2)=3 P(X=3)$,then the mean of $X$ is:

$A$ bag contains $2$ white and $1$ red balls. One ball is drawn at random and then put back in the box after noting its colour. The process is repeated again. If $X$ denotes the number of red balls recorded in the two draws,describe $X$.

$A$ random variable $X$ has the following probability distribution. For events $E = \{X \text{ is a prime number}\}$ and $F = \{X < 4\}$,what is the probability $P(E \cup F)$?

$X$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
$P(X)$	$0.15$	$0.23$	$0.12$	$0.10$	$0.20$	$0.08$	$0.07$	$0.05$

Given below is the probability distribution of a discrete random variable $X$:

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

Then $P(X \geq 4) = $

$A$ random variable $X$ has the range $\{0, 1, 2, \ldots\}$. If $P(X=r) = k(1+r) 3^{-r}$ for $r=0, 1, 2, \ldots$, where $k > 0$ is a real number, then $P(X=0) + P(X=1) + P(X=2) =$

It is known that a box of $8$ batteries contains $3$ defective pieces and a person randomly selects two batteries from the box. If $X$ is the number of defective batteries selected,then $P(X \leq 1) = $