4 The discrete random variable \(X\) has probability distribution defined by $$\mathrm { P } ( X = r ) = k ( 2 r - 1 ) \quad \text { for } r = 1,2,3,4,5,6 \text {, where } k \text { is a constant. }$$

1. Complete the table in the Printed Answer Booklet giving the probabilities in terms of \(k\).

   \(r\)	1	2	3	4	5	6
   \(\mathrm { P } ( X = r )\)

2. Show that the value of \(k\) is \(\frac { 1 } { 36 }\).
3. Draw a graph to illustrate the distribution.
4. In this question you must show detailed reasoning. Find
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\).
   A game consists of a player throwing two fair dice. The score is the maximum of the two values showing on the dice.
5. Show that the probability of a score of 3 is \(\frac { 5 } { 36 }\).
6. Show that the probability distribution for the score in the game is the same as the probability distribution of the random variable \(X\).
7. The game is played three times. Find
   • the mean of the total of the three scores.
   • the variance of the total of the three scores.