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Question 9.EX.30: Compute the expected system lifetime of a three-out-of-four ......

Compute the expected system lifetime of a three-out-of-four system when the first two component lifetimes are uniform on (0, 1) and the second two are uniform on (0, 2).

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r(p)=p1p2p3+p1p2p4+p1p3p4+p2p3p4−3p1p2p3p4r(1−F(t))={2(1−t)2(1−t/2)+2(1−t)(1−t/2)2−3(1−t)2(1−t/2)2,0⩽t⩽10,1⩽t⩽2E[ lifetime ]=∫01[2(1−t)2(1−t/2)+2(1−t)(1−t/2)2−3(1−t)2(1−t/2)2]dt=3160\begin{aligned}r({p})=p_1 p_2 p_3+ & p_1 p_2 p_4+p_1 p_3 p_4+p_2 p_3 p_4-3 p_1 p_2 p_3 p_4 \\r(\mathrm{1}-{F}(t))= & \left\{\begin{array}{lc}2(1-t)^2(1-t / 2)+2(1-t)(1-t / 2)^2 \\-3(1-t)^2(1-t / 2)^2, & 0 \leqslant t \leqslant 1 \\0, & 1 \leqslant t \leqslant 2\end{array}\right. \\E[\text { lifetime }]= & \int_0^1\left[2(1-t)^2(1-t / 2)+2(1-t)(1-t / 2)^2\right. \\& \left.-3(1-t)^2(1-t / 2)^2\right] d t \\= & \frac{31}{60}\end{aligned}

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