CAIE A-Level Further Math 4.1.3 PDF & CDF

Practise moving between PDFs and CDFs to calculate probabilities, fully defined functions, medians and percentiles.

Exam points

integrate a piecewise PDF to obtain F(x), including constants and interval domains
solve F(m)=0.5 or F(x)=p in the correct interval for medians and percentiles

[Maximum number: 2]

The continuous random variable X has probability density function f given by

f(x)= \begin{cases}0 & x<0 \\ \frac{6}{5} x & 0 \leqslant x \leqslant 1 \\ \frac{6}{5} x^{-4} & x>1\end{cases}

Find the median value of X.

Median satisfies $\frac{1}{2} \times m \times \frac{6}{5} m=\frac{1}{2}$

OR equivalent use of integration

1

M1

$m=\sqrt{\frac{5}{6}}$

1

A1

Correct value; accept 0.913

2

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4.1.3—PDF and CDF question 1