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 the probability that the train is more than 10 minutes late on each of two randomly chosen days. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
Exam No：9709_s21_qp_63 Year：2021 Question No：6(a)
Answer:
\(\mathrm{P}(X>10)=\int_{10}^{20} \frac{3}{8000}(x-20)^{2} \mathrm{~d} x\)
\(=\left[\frac{3}{8000} \times \frac{(x-20)^{3}}{3}\right]_{10}^{20}\) or \(\frac{3}{8000}\left[\frac{x^{3}}{3}-\frac{40 x^{2}}{2}+400 x\right]_{10}^{20}\)
\(=\frac{1}{8000}\left[0-(-10)^{3}\right]\)
\(\frac{1}{8}\) or \(0.125\)
\(\left(\frac{1}{8}\right)^{2}=\frac{1}{64}\) or \(0.0156(3 \mathrm{sf})\)
\(=\left[\frac{3}{8000} \times \frac{(x-20)^{3}}{3}\right]_{10}^{20}\) or \(\frac{3}{8000}\left[\frac{x^{3}}{3}-\frac{40 x^{2}}{2}+400 x\right]_{10}^{20}\)
\(=\frac{1}{8000}\left[0-(-10)^{3}\right]\)
\(\frac{1}{8}\) or \(0.125\)
\(\left(\frac{1}{8}\right)^{2}=\frac{1}{64}\) or \(0.0156(3 \mathrm{sf})\)
Knowledge points:
5.4.2 use formulae for probabilities for the binomial and geometric distributions, and recognise practical situations where these distributions are suitable models (Including the notations B(n,p) and Geo(p). Geo(p) denotes the distribution in which for r = 1,2,3,...)
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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