A2 June 2022 Q2

EdexcelCurrent spec9 marksDRVs

2. The discrete random variable \(X\) has probability distribution

\(x\)	\(-5\)	\(-1\)	0	5	\(b\)
\(\mathrm{P}(X = x)\)	0.3	0.25	0.1	0.15	0.2

where \(b\) is a constant and \(b \gt 5\)

(a) Find \(\mathrm{E}(X)\) in terms of \(b\) (1)

Given that \(\mathrm{Var}(X) = 34.26\)

(b) find the value of \(b\) (4)

Scheme	Marks	AO
[\(\mathrm{E}(X) =\)] \(0.2b - 1\)	B1	1.1b
	(1)

Mark scheme (b)

Scheme	Marks	AO
\(\mathrm{E}(X^2) = 25 \times 0.3 + 1 \times 0.25\,[+ 0 \times 0.1] + 25 \times 0.15 + 0.2b^2\ [= 11.5 + 0.2b^2]\)	M1	1.1b
\(\text{“}11.5 + 0.2b^2\text{”} - (\text{“}0.2b - 1\text{”})^2\ [= 34.26]\)	M1	3.1a
\(0.16b^2 + 0.4b - 23.76\ [= 0]\) or \(\tfrac{4}{25}b^2 + \tfrac{2}{5}b - \tfrac{594}{25}\ [= 0]\)	M1	1.1b
\(b = \underline{11}\) [since \(b \gt 5\)]	A1	2.2a
	(4)

Notes

1^st M1 Correct attempt at \(\mathrm{E}(X^2)\) using \(\sum x^2\mathrm{P}(X = x)\) at least 3 correct non-zero products
Allow \((-5)^2\) etc

2^nd M1 Realising that \(\mathrm{Var}(X) = \mathrm{E}(X^2) - [\mathrm{E}(X)]^2\) needs to be used

3^rd M1 Reducing their equation to a 3 term quadratic. At least 2 terms correct.
Allow e.g. \(0.16b^2 + 0.4b = 23.76\) Condone missing “=0”

A1 For 11 only (from the correct equation) so \(-13.5\) must be eliminated
Correct answer with no incorrect working seen scores 4/4

Mark scheme (c)

Scheme

Marks

\(X\)	\(-5\)	\(-1\)	\(0\)	\(5\)	\(\text{“}11\text{”}\)
\(X^2\)	\(25\)	\(1\)	\(0\)	\(25\)	\(\text{“}121\text{”}\)
\(2 - 3X\)	\(17\)	\(5\)	\(2\)	\(-13\)	\(\text{“}{-31}\text{”}\)
\(X^2 - 2\)	\(23\)	\(-1\)	\(-2\)	\(23\)	\(\text{“}119\text{”}\)
\(-3X\)	\(15\)	\(3\)	\(0\)	\(15\)	\(\text{“}{-33}\text{”}\)
\(X^2 + 3X\)	\(10\)	\(-2\)	\(0\)	\(40\)	\(\text{“}154\text{”}\)
\(X^2 + 3X - 2\)	\(8\)	\(-4\)	\(-2\)	\(38\)	\(\text{“}152\text{”}\)

M1
A1ft

2.1
1.1b

\(\mathrm{P}(X^2 \lt 2 - 3X) = \mathrm{P}(X = -1) + \mathrm{P}(X = 0)\)

2.2a

\(= \underline{0.35}\)

1.1b

(4)

Total 9

Notes

1^st M1 At least 4 values correct for (\(X^2\) and \(2 - 3X\)) or for (\(X^2 - 2\) and \(-3X\)) or \(X^2 + 3X\) or \(X^2 + 3X - 2\) (o.e.) Allow for solving equation with one sign error

1^st A1ft All correct or correct ft with their \(b\) but must have \(b \gt 5\) (accurate to 1 sf)
Allow solving equation to get awrt \(-3.6\) and awrt 0.56 or \(\frac{-3 \pm \sqrt{17}}{2}\) (ft their \(b \gt 5\))
If there are omissions but no errors in the lists of values then if 2^nd M1 and 2^nd A1 are scored then the 1^st M1 and 1^st A1 can be given by implication.

2^nd M1 For identifying the correct values of \(X\) required i.e. \(X = -1\) and \(X = 0\)

2^nd A1 0.35
NB It is possible to score M0A0M1A1 here if their table of values is incorrect
Correct answer with no incorrect working seen scores 4/4
(Allow correct use of their \(b \gt 5\))