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

$X$	$1$	$2$	$3$	$4$	$5$	$6$	$7$
$P(X)$	$k-1$	$3k$	$k$	$3k$	$3k^2$	$k^2$	$k^2+k$

Then the value of $k$ is:

$A$ random variable $X$ assumes values $1, 2, 3, \ldots, n$ with equal probabilities. If $\operatorname{var}(X) : E(X) = 4 : 1$,then $n$ is equal to

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

$X$	$0$	$1$	$2$	$3$	$4$
$P(X=x)$	$2k$	$k$	$2k$	$4k$	$k$

If $a = P(X < 3)$ and $b = P(2 < X < 4)$,then:

Bismuth has a half-life of $5$ days. If a sample originally has a mass of $800 \text{ mg}$, then the mass remaining after $30$ days will be: (in $\text{ mg}$)

The p.d.f. of a continuous random variable $X$ is given by $f(x) = \frac{1}{2}$ if $0 < x < 2$ and $f(x) = 0$ otherwise. If $a = P(X < \frac{1}{2})$ and $b = P(X > \frac{3}{2})$,then the relation between $a$ and $b$ is:

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

$X$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$
$P(X)$	$0$	$k$	$2k$	$3k$	$3k^2$	$k^2$	$2k^2$	$7k^2+k$

Determine $P(0 < X < 3)$. (in $/10$)