The waiting time, $$\(T\)$$ minutes, of a customer to be served in a local post office has probability density function $$\[ \mathrm{f}(t)= \begin{cases}\frac{1}{50}(18-2 t) & 0 \leqslant t \leqslant 3 \\ \frac{1}{20} & 3< t \leqslant 5 \\ 0 & \text { otherwise }\end{cases} \]$$ Given that the mean number of minutes a customer waits to be served is 1.66 (a) use algebraic integration to find $$\(\operatorname{Var}(T)\)$$, giving your answer to 3 significant figures. (5) (b) Find the cumulative distribution function $$\(\mathrm{F}(t)\)$$ for all values of $$\(t\)$$. (4) (c) Calculate the probability that a randomly chosen customer's waiting time will be more than 2 minutes. (2) (d) Calculate $$\(\mathrm{P}([\mathrm{E}(T)-2]< T< [\mathrm{E}(T)+2])\)$$ (2)

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