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

$X = x$	$0$	$1$	$2$
$P(X = x)$	$\frac{1}{5}$	$\frac{2}{5}$	$\frac{2}{5}$

Then the variance of $X$ is

If $X$ is a random variable with probability distribution $P(X=k) = \frac{(k+1)c}{2^k}$ for $k = 0, 1, 2, \ldots$,then $P(X \geq 3) = $

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 probability distribution of a random variable $X$ is given below. Then, the standard deviation of $X$ is

$X=x_i$	$2$	$3$	$5$	$7$	$12$
$P(X=x_i)$	$3k$	$k$	$k$	$2k$	$k$

Let $X$ be a random variable,and let $P(X=x)$ denote the probability that $X$ takes the value $x$. Suppose that the points $(x, P(X=x))$ for $x=0,1,2,3,4$ lie on a fixed straight line in the $xy$-plane,and $P(X=x)=0$ for all $x \in \mathbb{R} \setminus \{0,1,2,3,4\}$. If the mean of $X$ is $\frac{5}{2}$,and the variance of $X$ is $\alpha$,then the value of $24\alpha$ is:

The probability function of a discrete random variable $X$ is given by $P(X=r)=K r^2$, where $r=-2,-1,0,1,2,3$ and $K$ is a constant. The sum of the variance of $X$ and the square of the mean of $X$ is