The random variable \(X\) has mean14 and standard deviation 5. The independent random variable \(Y\) has mean 12 and standard deviation 3. The random variable \(W\) is given by \(W = X Y\). Find the value of
\(\quad \mathrm { E } ( W )\),
\(\quad \operatorname { Var } ( W )\).
The queueing times, \(T\) minutes, of customers at a local Post Office are modelled by the probability density function
$$\begin{array} { l l }
f ( t ) = \frac { 1 } { 2500 } t \left( 100 - t ^ { 2 } \right) & \text { for } 0 \leq t \leq 10
f ( t ) = 0 & \text { otherwise. }
\end{array}$$
Determine the mean queueing time.
Find the cumulative distribution function, \(F ( t )\), of \(T\).
Find the probability that a randomly chosen customer queues for more than 5 minutes.