3 Jeremy is a computing consultant who sometimes works at home. The number, \(X\), of days that Jeremy works at home in any given week is modelled by the probability distribution $$\mathrm { P } ( X = r ) = \frac { 1 } { 40 } r ( r + 1 ) \quad \text { for } r = 1,2,3,4 .$$

1. Verify that \(\mathrm { P } ( X = 4 ) = \frac { 1 } { 2 }\).
2. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
3. Jeremy works for 45 weeks each year. Find the expected number of weeks during which he works at home for exactly 2 days.