If the probability that the random variable $X$ takes the value $x$ is given by $P(X=x) = k(x+1)3^{-x}$,for $x = 0, 1, 2, 3, \ldots$,where $k$ is a constant,then $P(X \geq 3)$ is equal to

The probability distribution of a random variable $X$ is as follows. Then the mean of $X$ is

$X = x_{i}$	$-2$	$-1$	$0$	$1$	$2$
$P(X = x_{i})$	$k^2 / 3$	$k^2$	$2k^2 / 3$	$k / 2$	$k / 2$

If $P(X = x) = 5r^x$,$x = 1, 2, 3, \dots$ is the probability mass function of a discrete random variable $X$,then $r = $

If the mean of the following probability distribution of a random variable $X$ is $\frac{46}{9}$,then the variance of the distribution is:

$X$	$0$	$2$	$4$	$6$	$8$
$P(X)$	$a$	$2a$	$a+b$	$2b$	$3b$

Given the probability density function: $f(x) = \begin{cases} 3(1 - 2x^2), & 0 < x < 1 \\ 0, & \text{otherwise} \end{cases}$ The probability $P\left(\frac{1}{4} < X < \frac{1}{3}\right)$ is given by: $P\left(\frac{1}{4} < X < \frac{1}{3}\right) = \int_{1/4}^{1/3} 3(1 - 2x^2) \, dx$

If $X$ is a Poisson variable representing the number of successes in $50$ trials such that $2 P(X=1) = 5 P(X=5) + 2 P(X=3)$,then the probability of getting success in one trial is