CAIE A-Level Further Math 4.1.4 PDF & CDF

Practise using CDFs for related variables, then converting transformed distribution functions into PDFs and comparisons.

Exam points

substitute the transformation into F(x) to form the CDF of Y with correct domains
differentiate G(y) to find the PDF of Y after a transformation
compare medians by evaluating G at the median of X or linking the two median equations

[Maximum number: 3]

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

f(x)= \begin{cases}\frac{1}{6}\left(x^{-\frac{1}{3}}-x^{-\frac{2}{3}}\right) & 1 \leqslant x \leqslant 27 \\ 0 & \text { otherwise }\end{cases}

Find the probability density function of Y.

$\mathrm{G}(y)=\frac{1}{4}\left(y^{2}-2 y+1\right)$

M1

Using $y=x^{\frac{1}{3}}$ in their F(x).

$\mathrm{g}(y)=\frac{1}{2}(y-1)$

A1

Correct, AEF, 0 otherwise not required.

For $1 \leqslant y \leqslant 3$

B1

Seen anywhere, correct variable.

3

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4.1.4—PDF and CDF question 1