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

Question

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.

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The Answer of 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.

Assume That a Certain Dog Sled Team Has Race Times f(x)={4567x3−x2<x<50 else f ( x ) = \left\{ \begin{array} { l l } \frac { 4 } { 567 } x ^ { 3 } - x & 2 < x < 5 \\0 & \text { else }\end{array} \right.f(x)={5674​x3−x0​2<x<5 else ​

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