7 Each week, a newsagent stocks 5 copies of the magazine Statistics Weekly. A regular customer always buys one copy. The demand for additional copies may be modelled by a Poisson distribution with mean 2. The number of copies sold in a week, \(X\), has the probability distribution shown in the table, where probabilities are stated correct to three decimal places.

1. Show that, correct to three decimal places, the values of \(a\) and \(b\) are 0.180 and 0.143 respectively.
2. Find the values of \(\mathrm { E } ( X )\) and \(\mathrm { E } \left( X ^ { 2 } \right)\), showing the calculations needed to obtain these values, and hence calculate the standard deviation of \(X\).
3. The newsagent makes a profit of \(\pounds 1\) on each copy of Statistics Weekly that is sold and loses 50 p on each copy that is not sold. Find the mean weekly profit for the newsagent from sales of this magazine.
4. Assuming that the weekly demand remains the same, show that the mean weekly profit from sales of Statistics Weekly will be greater if the newsagent stocks only 4 copies.
   [0pt] [5 marks]
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