The range of a random variable $X$ is $\{1, 2, 3, \ldots\}$ and $P(X=x) = \frac{c^x}{x!}$ for $x = 1, 2, 3, \ldots$. Then the value of $c$ is

$A$ random variable $X$ has the following probability distribution. Then,$P(2 \leq X < 5) = $ . . . . . .

$X = x$	$1$	$2$	$3$	$4$	$5$	$6$
$P(X = x)$	$K$	$3K$	$5K$	$7K$	$8K$	$K$

The probability distribution of a discrete random variable $X$ is given by the following table:

$X$	$1$	$2$	$3$	$4$	$5$	$6$
$P(X)$	$K$	$2K$	$3K$	$4K$	$5K$	$6K$

Find the value of $P(2 < X < 6)$.

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

$X = x$	$-1$	$0$	$1$	$2$
$P(X = x)$	$\frac{1}{3}$	$\frac{1}{6}$	$\frac{1}{6}$	$\frac{1}{3}$

Then the value of $6 \Sigma(x^2) P(X=x) - \operatorname{var}(X) =$ ?

In a meeting,$70 \%$ of the members favour and $30 \%$ oppose a certain proposal. $A$ member is selected at random. We take $X=0$ if he opposes the proposal and $X=1$ if the member is in favour. Then the variance of $X$ is:

Suppose that two cards are drawn at random from a deck of cards. Let $X$ be the number of aces obtained. Then the value of $E(X)$ is