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STEP I 1997 question 14 solutionTSR Wiki > Study Help > Subjects and Revision > Mathematics > STEP > STEP 1997 Solutions > STEP I 1997 question 14 solution f(x) and T(x) are the probability (density) of the maximum height being x, and the cost if the maximum heigh becomes x, respectively. The expected cost, say, C, is then given by
(Note that
Differentiating:
which is strictly increasing. Setting
we get the minimum at
as required. If
will always be positive for y>=0. The expected cost is thus increasing for y >= 0, and so the lowest cost they can get is at y = 0. Thus the authorities shouldn't prepare at all when r + s < 1. Solution by ukgea |
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![\displaystyle C = \int_0^\infty ye^{-x}dx + \int_y^\infty e^{-x}(r - sy)dx + \left[-sxe^{-x}\right]_y^\infty + \int_y^\infty se^{-x}dx](http://www.thestudentroom.co.uk/latexrender/pictures/86/86e6ba5bf30d1b51945116fd477beb16.png "\displaystyle C = \int_0^\infty ye^{-x}dx + \int_y^\infty e^{-x}(r - sy)dx + \left[-sxe^{-x}\right]_y^\infty + \int_y^\infty se^{-x}dx")
![\displaystyle C = \left[-ye^{-x}\right]_0^\infty + \left[-(r-sy)e^{-x}\right]_y^\infty + \left[-sxe^{-x}\right]_y^\infty + \left[-se^{-x}\right]_y^\infty](http://www.thestudentroom.co.uk/latexrender/pictures/8b/8b7edd399b9daefe9b469bcb0e8cd4f9.png "\displaystyle C = \left[-ye^{-x}\right]_0^\infty + \left[-(r-sy)e^{-x}\right]_y^\infty + \left[-sxe^{-x}\right]_y^\infty + \left[-se^{-x}\right]_y^\infty")
