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 figures. $$\[ f(x)= \begin{cases}\frac{3}{8000}(x-20)^{2} & 0 \leqslant x \leqslant 20 \\ 0 & \text { otherwise. }\end{cases} \]$$ The median of $$$X$$$ is denoted by $$$m$$$. Show that $$$m$$$ satisfies the equation $$$(m-20)^{3}=-4000$$$, and hence find $$$m$$$ correct to 3 significant ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mathematics

IGCSE&ALevel

CAIE

Exam No：9709_s21_qp_63 Year：2021 Question No：6(c)

Answer:

$\int_{0}^{m} \frac{3}{8000}(x-20)^{2} d x=0.5$
$ {\left[\frac{3}{8000} \times \frac{(x-20)^{3}}{3}\right]_{0}^{m}=0.5\quad or\quad \frac{3}{8000}\left[\frac{x^{3}}{3}-\frac{40 x^{2}}{2}+400 x\right]_{0}^{m}=0.5 } $
$\frac{1}{8000}\left[(m-20)^{3}-(-20)^{3}\right]=0.5$
$(m-20)^{3}=-4000$
$(m=20+\sqrt[3]{-4000})$
$m=4.13(3 \mathrm{sf})$

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.)

Answer:

Knowledge points:

Solution:

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