4 A house has a total of five bedrooms, at least one of which is always rented.
The probability distribution for \(R\), the number of bedrooms that are rented at any given time, is given by $$\mathrm { P } ( R = r ) = \begin{cases} 0.5 & r = 1
0.4 ( 0.6 ) ^ { r - 1 } & r = 2,3,4
0.0296 & r = 5 \end{cases}$$

1. Complete the table below.
2. Find the probability that fewer than 3 bedrooms are not rented at any given time.
3. 1. Find the value of \(\mathrm { E } ( R )\).
   2. Show that \(\mathrm { E } \left( R ^ { 2 } \right) = 4.8784\) and hence find the value of \(\operatorname { Var } ( R )\).
4. Bedrooms are rented on a monthly basis. The monthly income, \(\pounds M\), from renting bedrooms in the house may be modelled by $$M = 1250 R - 282$$ Find the mean and the standard deviation of \(M\).

   \(\boldsymbol { r }\)	1	2	3	4	5
   \(\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )\)	0.5				0.0296