CAIE A-Level Further Math A2 4.1.2 Expected Values of Functions Questions

Practise calculating expectations, moments and variances of functions of continuous random variables with Further Mathematics Paper 2 questions and mark schemes.

Exam points

Apply de Moivre's theorem to powers and roots of complex numbers.

[Maximum number: 2]

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

\mathrm{f}(x)= \begin{cases}\frac{3}{28}\left(\mathrm{e}^{\frac{1}{2} x}+4 \mathrm{e}^{-\frac{1}{2} x}\right) & 0 \leqslant x \leqslant 2 \ln 3 \\ 0 & \text { otherwise }\end{cases}

Find $\mathrm{E}\left(Y^{4}\right)$ .

$\mathrm{E}\left(Y^{4}\right)=\int_{1}^{3} \frac{3}{28}\left(2+\frac{8}{y^{2}}\right) y^{4} \mathrm{~d} y=\int_{1}^{3} \frac{3}{28}\left(2 y^{4}+8 y^{2}\right) \mathrm{d} y=\frac{3}{28}\left[\frac{2}{5} y^{5}+\frac{8}{3} y^{3}\right]$

1

Integrate $y^{4} \times$ their g(y). Limits must be 1 and 3.

17.8

A1

$\frac{89}{5}$

2

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CAIE A-Level Further Math A2 4.1.2 Expected Values of Functions Questions question 1