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STEP II 2007 question 14 solution
From The Student RoomTSR Wiki > Study Help > Subjects and Revision > Mathematics > STEP > STEP II 2007 question 14 solution (i) The sketch should be straightforward: a curve (in the shape of (ii) Noticing the less than [B]or equal to[/B] signs, we can say that
Solving these simultaneously yields:
Now the total area under the pdf must be equal to 1, and hence
Evaluating the integral and gathering like terms, we find
Now as
(iii) Using as estimate for Let us call the median
Evaluating the integral and tidying up, we see
Solution by coffeym. |










) between x=1 and x=k, a straight horizontal line up until x=2k, and then a negative gradient straight line which reaches the x-axis at x=4k
(by substituting x=2k into the pdf) and also
as required for the first part.
(note the areas of two of the sections are simply rectangles and triangles and so integration is overkill)
we can say that
, and by the relations we derived earlier,
close to 1, we can see that the median lies in the range
, then using the idea of rectangles and triangles under our pdf as before we have:
by the definition of the median that:
as required




