5. A piece of wood \(A B\) is 3 metres long. The wood is cut at random at a point \(C\) and the random variable \(W\) represents the length of the piece of wood \(A C\).
- Find the probability that the length of the piece of wood \(A C\) is more than 1.8 metres.
The two pieces of wood \(A C\) and \(C B\) form the two shortest sides of a right-angled triangle. The random variable \(X\) represents the length of the longest side of the right-angled triangle.
- Show that \(X ^ { 2 } = 2 W ^ { 2 } - 6 W + 9\)
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[You may assume for random variables \(S , T\) and \(U\) and for constants \(a\) and \(b\) that if \(S = a T + b U\) then \(\mathrm { E } ( S ) = a \mathrm { E } ( T ) + b \mathrm { E } ( U ) ]\) - Find \(\mathrm { E } \left( X ^ { 2 } \right)\)
- Find \(\mathrm { P } \left( X ^ { 2 } > 5 \right)\)