7. A piece of string \(A B\) has length 9 cm . The string is cut at random at a point \(P\) and the random variable \(X\) represents the length of the piece of string \(A P\).
- Write down the distribution of \(X\).
- Find the probability that the length of the piece of string \(A P\) is more than 6 cm .
The two pieces of string \(A P\) and \(P B\) are used to form two sides of a rectangle.
The random variable \(R\) represents the area of the rectangle.
- Show that \(R = a X ^ { 2 } + b X\) and state the values of the constants \(a\) and \(b\).
- Find \(\mathrm { E } ( R )\).
- Find the probability that \(R\) is more than twice the area of a square whose side has the length of the piece of string \(A P\).