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 $E\left(X\right)$.

Complete this probability distribution, in terms of $k$, for a discrete random variable $X$

What range of values can $k$ take? Give your answer in the form $a\le k\le b, a, b\in Q$

Complete this probability distribution, in terms of $k$, for a discrete random variable $X$

Find the mean of the distribution in terms of $k$.

Ten balls of identical size are in a bag. Two of the balls are red and the rest are blue. Balls are picked out at random from the bag and are not replaced. Let $R$ be the number of balls drawn out, up to and including the first red one. List the possible values of $R$ and their associated probabilities.

Ten balls of identical size are in a bag. Two of the balls are red and the rest are blue. Balls are picked out at random from the bag and are not replaced. Let $R$ be the number of balls drawn out, up to and including the first red one. Calculate the mean value of $R$.

Ten balls of identical size are in a bag. Two of the balls are red and the rest are blue. Balls are picked out at random from the bag and are not replaced. Let $R$ be the number of balls drawn out, up to and including the first red one. What is the most likely value of $R$ ?

Consider the bag of balls in question $8$ Suppose now that each ball is replaced before the next is drawn. Show that the probability of the first red being drawn out on the second go is $\frac{4}{25}$.