Alethia models the length of time, in minutes, by which her train is late on any day by the random variable $$\(X\)$$ with probability density function given by $$\( f(x)= \begin{cases}\frac{3}{8000}(x-20)^{2} & 0 \leqslant x \leqslant 20 \\ 0 & \text { otherwise. }\end{cases} \)$$ Find $$\(\mathrm{E}(X)\)$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
Exam No：9709_s21_qp_63 Year：2021 Question No：6(b)
Answer:
\(\int_{0}^{20} \frac{3}{8000}\left(x^{3}-40 x^{2}+400 x\right) \mathrm{d} x\)
\(\frac{3}{8000}\left[\frac{x^{4}}{4}-\frac{40 x^{3}}{3}+\frac{400 x^{2}}{2}\right]_{0}^{20}\)
or \(\left(\frac{3 x}{8000} \times \frac{(x-20)^{3}}{3}\right)-\frac{1}{8000}\left(\frac{(x-20)^{4}}{4}\right)\)
\(\frac{3}{8000}\left[\frac{160000}{4}-\frac{40 \times 8000}{3}+200 \times 400\right]\)
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\(\frac{3}{8000}\left[\frac{x^{4}}{4}-\frac{40 x^{3}}{3}+\frac{400 x^{2}}{2}\right]_{0}^{20}\)
or \(\left(\frac{3 x}{8000} \times \frac{(x-20)^{3}}{3}\right)-\frac{1}{8000}\left(\frac{(x-20)^{4}}{4}\right)\)
\(\frac{3}{8000}\left[\frac{160000}{4}-\frac{40 \times 8000}{3}+200 \times 400\right]\)
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Knowledge points:
6.3.2 use a probability density function to solve problems involving probabilities, and to calculate the mean and variance of a distribution. (Including location of the median or other percentiles of a distribution by direct consideration of an area using the density function.) (Explicit knowledge of the cumulative distribution function is not included.)
Solution:
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