CAIE A-Level Further Mathematics 4.1 Continuous Random Variables Question Bank

Practise piecewise density and distribution functions, transformed variables, expectations, variances, medians and percentiles.

[Maximum number: 8]

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}

(a)

Find P(X>1).

[ 1 ]

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

OR

1

B1

(b)

Find the median value of X.

[ 2 ]

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

(c)

Given that E(X)=1, find the variance of X.

[ 3 ]

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

1

M1

Attempt evaluation of the sum of these two
integrals

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

1

A1

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

(d)

Find $\mathrm{E}(\sqrt{X})$ .

[ 2 ]

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

1

M1

Attempt evaluation of the sum of these two
integrals

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

1

A1

2

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CAIE A-Level Further Mathematics 4.1 Continuous Random Variables Question Bank

Exam points

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Covered syllabus points

4.1 Continuous random variables question 1

Question (a)

Question (b)

Question (c)

Question (d)