Assume That a Certain Dog Sled Team Has Race Times $$f ( x ) = \left\{ \begin{array} { l l } \frac { 4 } { 567 } x ^ { 3 } - x & 2 < x < 5 \\0 & \text { else }\end{array} \right.$$

Short Answer

Assume that a certain dog sled team has race times, in hours, which are a random variable with probability density function $$f ( x ) = \left\{ \begin{array} { l l } \frac { 4 } { 567 } x ^ { 3 } - x & 2 < x < 5 \\0 & \text { else }\end{array} \right.$$
(a) Find the expected duration of such a race.
(b) Find the probability that if the dogs run 5 races, then none of them is shorter than 3 hours. Assume the duration of the individual races are independent of one another.