CAIE A-Level Further Mathematics 4.1.1 Piecewise Density Functions

Practise normalising a density defined over several intervals, sketching its joined pieces and integrating the correct branch to find probabilities or quantiles.

Exam points

integrate every density branch over its stated interval and set their total area equal to one
sketch each piece with correct endpoints, joins and zero density outside the support
select the appropriate branch of the cumulative integral when finding a probability or median

[Maximum number: 1]

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 P(X>1).

$\mathrm{P}(X>1)=1-\frac{1}{2} \times 1 \times \frac{6}{5}=\frac{2}{5}$

OR

1

B1

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4.1.1—Piecewise density functions question 1