Pearson Edexcel IAL Mathematics S2.2.5 Mode, median & quartiles of continuous random variables

Practise locating modes, medians, quartiles and percentiles for continuous variables from density functions and cumulative probabilities.

Exam points

solve F(x)=0.5 or F(x)=p to find medians, quartiles or percentiles
use derivative tests or the shape of f(x) to find the mode of a continuous variable
compare F(a) and F(b) with 0.5 to justify an interval containing the median

[Maximum number: 5]

A random variable X has probability density function given by

f(x)=\left\{\begin{array}{cc} \frac{1}{4} & -\frac{1}{2} \leqslant x<\frac{1}{2} \\ 2 x-\frac{3}{4} & \frac{1}{2} \leqslant x \leqslant k \\ 0 & \text { otherwise } \end{array}\right.

where k is a positive constant.

Calculate the interquartile range of X

$Q_1=0.5$ $Q_3^2-\frac{3}{4}Q_3+0.125=0.5$ $Q_3=\frac{\frac34+\sqrt{\left(-\frac34\right)^2+4\left(\frac18\right)}}{2}=1.093\ldots$ $\operatorname{IQR}=1.093\ldots-0.5=0.593\ldots$ $\operatorname{IQR}=\text{awrt }0.59$
B1 for 0.5; M1 for equation for $Q_3$ ; M1 for solving; M1 for IQR; A1 for awrt 0.59.

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S2.2.5 - Mode, median and quartiles of continuous random variables question 1