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The duration $Y$ of long-distance telephone calls(in minutes) monitored by a station is a random variable with the properties that $P(Y=3)=0.2$ and $P(Y=6)=0.1$ Otherwise,Y has a continuous density function given by $$ f(y)= \begin{cases} \tfrac{1}{4}ye^{-y/2},\quad y>0,\\ 0,\qquad\qquad\text{elsewhere}. \end{cases} $$

The discrete points at 3 and 6 are due to the fact that the length of the call is announced to the called in three minute intervals and the caller must pay for three minutes even if he talks less than three minutes. Find the expected duration of a randomly selected long distance call.

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Hint: Write $Y=Y\mathbf{1}_{\{0<Y<3\}}+3\mathbf{1}_{\{Y=3\}}+Y\mathbf{1}_{\{3<Y<6\}}+6\mathbf{1}_{\{Y=6\}}+Y\mathbf{1}_{\{Y>6\}}$.

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