Let t be the time, in hours, it takes for a student to complete a ?nal exam. All

Question

Let t be the time, in hours, it takes for a student to complete a ?nal exam.
All students complete the exam within two [one] hours and the density function for t
is given by:
p(t) = (1/b)(4t ? t^3) if 0 < t < 2
0 otherwise

a) Find the value of b
b) What is the mean time for students to complete the ?nal exam? Express your answer (i) symbolically in terms of b and (ii) numerically, using the value of b you found in part (a).
c) Determine the cumulative distribution function P(t). Express your answer (i)
symbolically in terms of b and (ii) numerically, using the value of b you found in
part (a).

Explanation / Answer

p(t) = (1/b)(4t - t^3) a) integral(pdt) =1 integral((1/b)(4t - t^3)dt) = 1 (limit t = 0 to 2) (1/b)(2*2^2 - 2^4/4) = 1 (1/b)*4 =1 b = 4 b) mean = integral(t*p)dt = integral((1/b)(4t^2 - t^4)dt (limit t= 0 to 2) = (1/b)((4*8/3) - (32/5)) =64/15b = 16/15 hrs c) cdf = integral(pdt) (limit t = 0 to t) = integral((1/b)(4t - t^3))dt = (1/b)*(2t^2 - 0.25t^4) for 0

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