AS October 2020 Q3

EdexcelAS paperCurrent spec14 marksContinuous Random Variables

3. The continuous random variable \(X\) has cumulative distribution function

\[\mathrm{F}(x) = \begin{cases} 0 & x \lt 4 \\ px - k\sqrt{x} & 4 \leqslant x \leqslant 9 \\ 1 & x \gt 9 \end{cases}\]

where \(p\) and \(k\) are constants.

(a) Find the value of \(p\) and the value of \(k\). (4)

Given that \(\mathrm{E}(X) = \dfrac{119}{18}\)

(b) show that \(\mathrm{Var}(X) = 2.05\) to 3 significant figures. (6)

(d) Find the exact value of the constant \(a\) such that \(\mathrm{P}(X \leqslant a) = \dfrac{7}{27}\) (3)

Mark scheme (a)

Scheme	Marks	AO
\(\mathrm{F}(4) = 0 \ \Rightarrow\ 4p - 2k = 0\) or \(\mathrm{F}(9) = 1 \ \Rightarrow\ 9p - 3k = 1\) Both	M1 A1	2.1 1.1b
Solving e.g. sub \(k = 2p \ \Rightarrow\ 9p - 6p = 1\)	M1	1.1b
\(\underline{p = \dfrac{1}{3} \quad k = \dfrac{2}{3}}\)	A1	1.1b
	(4)

1^st M1 for selecting a correct approach and getting 1 correct equation

1^st A1 for 2 correct equations in \(p\) and \(k\)

2^nd M1 for solving two simultaneous equations (based on \(\mathrm{F}(x)\))– reducing to a linear eqn in 1 var

2^nd A1 for both correct values

Mark scheme (b)

Scheme	Marks	AO
Find \(\mathrm{f}(x)\): \(\mathrm{f}(x) = \mathrm{F}'(x) = \text{“}\tfrac{1}{3}\text{”} - \text{“}\tfrac{2}{3}\text{”} \times \dfrac{1}{2}x^{-\frac{1}{2}} = \dfrac{1}{3}\left(1 - x^{-\frac{1}{2}}\right)\)	M1	3.1a
\(\mathrm{E}\left(X^2\right) = \dfrac{1}{3}\displaystyle\int_4^9 x^2\left(1 - x^{-\frac{1}{2}}\right)\mathrm{d}x\) or \(\dfrac{1}{3}\displaystyle\int_4^9 \left(x^2 - x^{\frac{3}{2}}\right)\mathrm{d}x\)	M1	2.1
\(= \dfrac{1}{3}\left[\dfrac{x^3}{3} - \dfrac{2x^{\frac{5}{2}}}{5}\right]_4^9\) or \(\dfrac{1}{3}\left[\left(\dfrac{9^3}{3} - \dfrac{2 \times 3^5}{5}\right) - \left(\dfrac{64}{3} - \dfrac{2 \times 2^5}{5}\right)\right] = \dfrac{2059}{45}\)	M1 A1	1.1b 1.1b
\(\mathrm{Var}(X) = \text{“}\dfrac{2059}{45}\text{”} - \left(\dfrac{119}{18}\right)^2\)	M1	1.1b
\(= 2.048765\ldots = 2.05\) (3sf) *()**	A1*	1.1b
	(6)

1^st M1 for realising need to find \(\mathrm{f}(x)\) first and attempt to differentiate \(\mathrm{F}(x)\) – some correct

2^nd M1 for attempting \(\displaystyle\int x^2\mathrm{f}(x)\,\mathrm{d}x\) ft their \(\mathrm{f}(x)\) provided different from \(\mathrm{F}(x)\)

3^rd M1 for some correct integration using their \(\mathrm{f}(x)\) or a numerical expression for \(\mathrm{E}(X^2)\)

1^st A1 for a correct value for \(\mathrm{E}(X^2)\) exact fraction or at least 45.755….

4^th M1 for a correct method for \(\mathrm{Var}(X)\) ft their \(\mathrm{E}(X^2)\)

2^nd A1* for a fully correct solution

Scheme	Marks	AO
[\(\mathrm{f}(x) = \dfrac{1}{3}\left(1 - \dfrac{1}{\sqrt{x}}\right)\) so max is when \(x\) is greatest so] mode = 9	B1	2.2a
	(1)

Mark scheme (d)

Scheme	Marks	AO
\(\mathrm{F}(a) = \dfrac{7}{27} \ \Rightarrow\ a - 2\sqrt{a} - \dfrac{7}{9} = 0\)	M1	3.1a
\(y^2 - 2y - \dfrac{7}{9} = 0 \ \Rightarrow\ (y - 1)^2 = \dfrac{16}{9}\)	M1	2.1
\(y = \dfrac{7}{3}\) or \(\left(-\dfrac{1}{3}\ \text{not valid}\right)\) so \(\underline{a = \dfrac{49}{9}}\)	A1	3.2a
	(3)
	(14 marks)

1^st M1 for realising the need to use \(\mathrm{F}(x)\) and forming a correct equation in \(a\) (ft their \(p\) and \(k\))

2^nd M1 for recognising equation as a quadratic and trying to solve

A1 for exact working and selection of the appropriate value