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 $C$.

When throwing a normal dice, let $X$ be the random variable defined by $X=$the square of the score shown on the dice. What is the expectation of $X$?

A "Fibonacci dice" is unbiased, six-sided and labelled with these numbers: $1, 2, 3, 5, 8, 13$. What is the expected score when the dice is rolled?

The discrete random variable X has probability distribution $P\left(x\right)=\frac{x}{36}$ for $x= 1, 2, 3,..... 8$. Find $E\left(X\right)$.

For the discrete random variable $X$, the probability distribution is given by

$P(X=x)=\left\{\begin{array}{ll}kx& x=1,2,3,4,5\\ k\left(10-x\right)& x=6,7,8,9\end{array}\right.$

Find the value of the constant $k$.

For the discrete random variable $X$, the probability distribution is given by

$P(X=x)=\left\{\begin{array}{ll}kx& x=1,2,3,4,5\\ k\left(10-x\right)& x=6,7,8,9\end{array}\right.$

Find $E\left(X\right)$.