The random variable $X$ has the probability distribution shown.

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

Find the value of $c$.

The random variable $X$ has the probability distribution shown.

Find $P(1 < X<4)$.

The probability distribution of a random variable $Y$ is given by: $P(Y= y) = c{y}^3$ for $y = 1, 2, 3$. Given that $c$ is a constant, find the value of $c$.

The random variable X has the probability distribution shown.

Find the value of $k$.

The random variable $X$ has the probability distribution given by $P(X=x)= k{\left(\frac{1}{3}\right)}^{x-1}$ for$x= 1, 2, 3, 4,$ where $k$ is a constant. Find the exact value of $k$.

The discrete random variable $X$ can take only the values $0, 1, 2, 3, 4, 5$. The probability distribution of $X$ is given by the following 

$P(X=0)=P(X=1)=P(X= 2) = a$

$P(X=3)=P(X=4)=P(X= 5) = b$

$P(X\ge 2)=3P(X<2)$

where $a$ and $b$ are constants.

Determine the values of $a$ and $b$.

The discrete random variable $X$ can take only the values $0, 1, 2, 3, 4, 5$. The probability distribution of $X$ is given by the following 

$P(X=0)=P(X=1)=P(X= 2) = a$

$P(X=3)=P(X=4)=P(X= 5) = b$

$P(X\ge 2)=3P(X<2)$

where $a$ and $b$ are constants.

Determine the probability that the sum of two independent observations from this distribution exceeds $7$.

The discrete random variables A and B are independent and have the following distributions.

The random variable $C$ is the sum of one observation from $A$ and one observation from $B$.

Show that $P(C=3)= \frac{5}{18}$.

The discrete random variables A and B are independent and have the following distributions.

The random variable $C$ is the sum of one observation from A and one observation from $B$. Tabulate the probability distribution for $C$.