If a discrete random variable $X$ takes values $0, 1, 2, 3, \ldots$ with probability $P(X=x) = k(x+1) 5^{-x}$,where $k$ is a constant,then $P(X=0)$ is

If the function $f$ defined by $f(x) = \begin{cases} K(x-x^2) & \text{if } 0 < x < 1 \\ 0 & \text{otherwise} \end{cases}$ is the probability density function (p.d.f.) of a random variable $X$,then the value of $P(X < \frac{1}{2})$ is

The distribution of a random variable $X$ is given below. The value of $k$ is:

$X = x$	$-2$	$-1$	$0$	$1$	$2$	$3$
$P(X = x)$	$\frac{1}{10}$	$k$	$\frac{1}{5}$	$2k$	$\frac{3}{10}$	$k$

$A$ random variable $X$ assumes values $1, 2, 3, \ldots, n$ with equal probabilities. If the ratio of the variance of $X$ to the expected value of $X$ is equal to $4$,then the value of $n$ is:

Find the probability distribution of the number of successes in two tosses of a die,where a success is defined as a number greater than $4$.

If the probability mass function (p.m.f.) of a discrete random variable $X$ is given by $P(X=x) = \frac{c}{x^3}$ for $x = 1, 2, 3$ and $0$ otherwise,then $E(X)$ is equal to: