Modelling assumptions and refinements

6. A discrete random variable is such that each of its values is assumed to be equally likely.

1. Write down the name of the distribution that could be used to model this random variable.
2. Give an example of such a distribution.
3. Comment on the assumption that each value is equally likely.
4. Suggest how you might refine the model in part (a).

The discrete random variable \(X\) can take any value in the set \(\{1, 2, 3, 4, 5, 6, 7, 8\}\). Arthur, Beatrice and Chris each carry out trials to investigate the distribution of \(X\). Arthur finds that P\((X = 1) = 0.125\) and that E\((X) = 4.5\). Beatrice finds that P\((X = 2) =\) P\((X = 3) =\) P\((X = 4) = p\). Chris finds that the values of \(X\) greater than 4 are all equally likely, with each having probability \(q\).

1. Calculate the values of \(p\) and \(q\). [7 marks]
2. Give the name for the distribution of \(X\). [1 mark]
3. Calculate the standard deviation of \(X\). [3 marks]

The random variable \(X\) represents the value on the upper face of an eight-sided dice after it has been rolled. The faces are numbered 1 to 8 The random variable \(X\) is modelled by a discrete uniform distribution with \(n = 8\)

1. Find E\((X)\) [1 mark]
2. Find Var\((X)\) [1 mark]
3. Find P\((X \geq 6)\) [1 mark]
4. The dice was rolled 800 times and the results below were obtained.

   \(x\)	1	2	3	4	5	6	7	8
   Frequency	103	63	84	110	74	41	85	240

   State, with a reason, how you would refine the model for the random variable \(X\). [2 marks]