Edexcel A-Level Mathematics A2 S2.2.5 Mode Median and Quartiles of Continuous Random Variables Questions

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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Edexcel A-Level Mathematics A2 S2.2.5 Mode Median and Quartiles of Continuous Random Variables Questions question 1