A2 June 2023 Q6

6. The continuous random variable \(X\) has cumulative distribution function given by

\[\mathrm{F}(x) = \begin{cases} 0 & x \lt 0 \\ k\left(x - ax^2\right) & 0 \leqslant x \leqslant 4 \\ 1 & x \gt 4 \end{cases}\]

The values of \(a\) and \(k\) are positive constants such that \(\mathrm{P}(X \lt 2) = \dfrac{2}{3}\)

(a) Find the exact value of the median of \(X\) (6)

(b) Find the probability density function of \(X\) (2)

Mark scheme (a)

Scheme	Marks	AO
\(\mathrm{F}(2) = \frac{2}{3} \quad\) and \(\quad \mathrm{F}(4) = 1\)
\(k(2 - 4a) = \frac{2}{3} \qquad k(4 - 16a) = 1\)	M1	2.1
\(\dfrac{2}{3(2 - 4a)} = \dfrac{1}{(4 - 16a)} \to a = \dfrac{1}{10},\quad k = \dfrac{5}{12}\)	M1A1	1.1b 1.1b
\(\frac{5}{12}\left(m - \frac{1}{10}m^2\right) = 0.5 \to m^2 - 10m + 12 = 0\)	M1M1	3.1a 1.1b
\(m = 5 - \sqrt{13} \qquad (m = 5 + \sqrt{13}\ \text{ reject})\)	A1	1.1b
	(6)

M1: using \(\mathrm{F}(2) = \frac{2}{3}\) and \(\mathrm{F}(4) = 1\) to form two correct expressions in \(a\) and \(k\)

M1: solving simultaneously to find a value of \(a\) or \(k\)

A1: both \(a = \dfrac{1}{10}\) and \(k = \dfrac{5}{12}\), may be implied by correct equation or median

M1: setting up \(\mathrm{F}(m) = 0.5\) to form a 3TQ

M1: using an appropriate method to solve quadratic, resulting in a value of \(m\)

A1: selecting the correct value for \(m\)

Mark scheme (b)

Scheme	Marks	AO
\(\mathrm{f}(x) = \dfrac{\mathrm{d}}{\mathrm{d}x}\big(\mathrm{F}(x)\big)\)	M1	1.1b
\(\mathrm{f}(x) = \begin{cases} \text{“}\tfrac{5}{12}\text{”}(1 - \text{“}0.2\text{”}x) & 0 \leqslant x \leqslant 4 \\ 0 & \text{otherwise} \end{cases}\)	A1ft	1.1b
	(2)

M1: differentiating \(\mathrm{F}(x)\), at least one term correct for their \(a\) and \(k\)

A1ft: correct ft expression with correct limits

Mark scheme (c)

Scheme	Marks	AO
Mode is \(X = 0\)…	B1	2.2a
…since \(\mathrm{f}(x)\) is linear with negative gradient. or …\(\mathrm{f}(x)\) is a decreasing function	dB1	2.4
	(2)
	(10 marks)

B1: deducing that the mode is \(X = 0\)

dB1: correct reasoning
May be given via a diagram of a linear function \(0 \leqslant x \leqslant 4\) with: