If the probability mass function (p.m.f.) of a random variable $X$ is given by the table below,then find the value of $k$.

$X$	$1$	$2$	$3$	$4$	$5$
$P(X = x)$	$k$	$\frac{k}{3}$	$\frac{k}{4}$	$\frac{k}{2}$	$\frac{k}{2}$

If $X$ is a Poisson variate with $P(X=0)=0.8$,then the variance of $X$ is

If the mean of a Poisson distribution is $\frac{1}{2}$,then the ratio of $P(X=3)$ to $P(X=2)$ is

Let $X$ denote the number of hours you study on a Sunday. It is known that $P(X=x) = \begin{cases} 0.1 & \text{if } x=0 \\ kx & \text{if } x=1, 2 \\ k(5-x) & \text{if } x=3, 4 \\ 0 & \text{otherwise} \end{cases}$ where $k$ is a constant. Then the probability that you study at least two hours on a Sunday is

The random variable $X$ has a probability distribution $P(X)$ of the following form,where $k$ is some number:
$P(X) = \begin{cases} k, & \text{if } x=0 \\ 2k, & \text{if } x=1 \\ 3k, & \text{if } x=2 \\ 0, & \text{otherwise} \end{cases}$
Determine the value of $k$.

$A$ random variable $X$ has the following probability distribution:

$x$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
$P(X=x)$	$k$	$2k$	$3k$	$4k$	$4k$	$3k$	$2k$	$k$	$k$

Then $P(3 < X \leq 6) = $