- (i) The continuous random variable \(X\) is uniformly distributed over the interval \([ a , b ]\)
Given that \(\mathrm { P } ( 5 < X < 13 ) = \frac { 1 } { 5 }\) and \(\mathrm { E } ( X ) = 9\), find \(\mathrm { P } ( 3 X > a + b )\)
(ii) The continuous random variable \(Y\) is uniformly distributed over the interval \([ 1 , c ]\) Given that \(\operatorname { Var } ( Y ) = 0.48\), find the exact value of \(\mathrm { E } \left( Y ^ { 2 } \right)\)
(iii) A wire of length 20 cm is cut into 2 pieces at a random point.
The longest piece of wire is then cut into 2 pieces, equal in length, giving 3 pieces of wire altogether.
Find the probability that the length of the shortest piece of wire is less than 6 cm .