5. The continuous random variable \(X\) is uniformly distributed over the interval \(\alpha < x < \beta\).
- Write down the probability density function of \(X\), for all \(x\).
- Given that \(\mathrm { E } ( X ) = 2\) and \(\mathrm { P } ( X < 3 ) = \frac { 5 } { 8 }\) find the value of \(\alpha\) and the value of \(\beta\).
A gardener has wire cutters and a piece of wire 150 cm long which has a ring attached at one end. The gardener cuts the wire, at a randomly chosen point, into 2 pieces. The length, in cm, of the piece of wire with the ring on it is represented by the random variable \(X\). Find
- \(\mathrm { E } ( X )\),
- the standard deviation of \(X\),
- the probability that the shorter piece of wire is at most 30 cm long.