CAIE A-Level Further Math A2 4.1 Continuous Random Variables Questions

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

OR

1

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

OR equivalent use of integration

1
$m=\sqrt{\frac{5}{6}}$

1
Correct value;
accept 0.913

2

$\mathrm{E}\left(X^{2}\right)=\int_{0}^{1} \frac{6}{5} x^{3} \mathrm{~d} x+\int_{1}^{\infty} \frac{6}{5} x^{-2} \mathrm{~d} x$

1
Attempt evaluation of the sum of these two
integrals

$\mathrm{E}\left(X^{2}\right)=1.5$

1
Correct value seen or implied

$\operatorname{Var}(X)=1.5-1^{2}=0.5$

1

A1FT

Follow through their 1.5 providing variance is
positive

3

$\mathrm{E}(\sqrt{X})=\int_{0}^{1} \frac{6}{5} x^{\frac{3}{2}} \mathrm{~d} x+\int_{1}^{\infty} \frac{6}{5} x^{-\frac{7}{2}} \mathrm{~d} x$

1
Attempt evaluation of the sum of these two
integrals

$\frac{24}{25}$ or 0.96

1
2