1 The probability distribution of a discrete random variable \(X\) is given by the formula \(\mathrm { P } ( \mathrm { X } = \mathrm { r } ) = \mathrm { k } \left( ( \mathrm { r } - 1 ) ^ { 2 } + 1 \right)\) for \(r = 1,2,3,4,5\).

1. Show that \(k = \frac { 1 } { 35 }\). The distribution of \(X\) is shown in the table.

   \(r\)	1	2	3	4	5
   \(\mathrm { P } ( \mathrm { X } = \mathrm { r } )\)	\(\frac { 1 } { 35 }\)	\(\frac { 2 } { 35 }\)	\(\frac { 1 } { 7 }\)	\(\frac { 2 } { 7 }\)	\(\frac { 17 } { 35 }\)

2. Comment briefly on the shape of the distribution.
3. Find each of the following.
   • \(\mathrm { E } ( X )\)
   • \(\operatorname { Var } ( X )\)
   The random variable \(Y\) is given by \(Y = 5 X - 10\).
4. Find each of the following.
   • \(\mathrm { E } ( Y )\)
   • \(\operatorname { Var } ( Y )\)