11 The time, \(T\) years, before a particular type of washing machine breaks down may be taken to have probability density function f given by
$$\mathrm { f } ( t ) = \begin{cases} a t \mathrm { e } ^ { - b t } & t > 0 \\ 0 & \text { otherwise } \end{cases}$$
where \(a\) and \(b\) are positive constants. It may be assumed that, if \(n\) is a positive integer,
$$\int _ { 0 } ^ { \infty } t ^ { n } \mathrm { e } ^ { - b t } \mathrm {~d} t = \frac { n ! } { b ^ { n + 1 } }$$
- Records show that the mean of \(T\) is 1.5 . Show that \(b = \frac { 4 } { 3 }\) and find the value of \(a\).
- Find \(\operatorname { Var } ( T )\).
- Calculate \(\mathrm { P } ( T < 1.5 )\). State, giving a reason, whether this value indicates that the median of \(T\) is smaller than the mean of \(T\) or greater than the mean of \(T\).