The three independent random variables \(A , B\) and \(C\) each have a continuous uniform distribution over the interval \([ 0,5 ]\).
Find the probability that \(A , B\) and \(C\) are all greater than 3
The random variable \(Y\) represents the maximum value of \(A , B\) and \(C\).
The cumulative distribution function of \(Y\) is
$$\mathrm { F } ( y ) = \begin{cases} 0 & y < 0 \frac { y ^ { 3 } } { 125 } & 0 \leqslant y \leqslant 5 1 & y > 5 \end{cases}$$
Using algebraic integration, show that \(\operatorname { Var } ( Y ) = 0.9375\)
Find the mode of \(Y\), giving a reason for your answer.
Describe the skewness of the distribution of \(Y\). Give a reason for your answer.
Find the value of \(k\) such that \(\mathrm { P } ( k < Y < 2 k ) = 0.189\)