6. The three independent random variables \(A , B\) and \(C\) each has a continuous uniform distribution over the interval \([ 0,5 ]\).
- Find \(\mathrm { P } ( A > 3 )\).
- 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
$$\mathrm { F } ( y ) = \begin{cases} 0 & y < 0
\frac { y ^ { 3 } } { 125 } & 0 \leqslant y \leqslant 5
1 & y > 5 \end{cases}$$ - Find the probability density function of \(Y\).
- Sketch the probability density function of \(Y\).
- Write down the mode of \(Y\).
- Find \(\mathrm { E } ( Y )\).
- Find \(\mathrm { P } ( Y > 3 )\).