AS June 2024 Q3

EdexcelAS paperCurrent spec6 marksDRVs

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

\(x\)	\(-1\)	0	1	3	7
\(\mathrm{P}(X = x)\)	\(p\)	\(r\)	\(p\)	0.3	\(r\)

where \(p\) and \(r\) are probabilities.

Given that \(\mathrm{E}(X) = 1.95\)

find the exact value of \(\mathrm{E}\left(\sqrt{X+1}\right)\) giving your answer in the form \(a + b\sqrt{2}\) where \(a\) and \(b\) are rational. (6)

Mark scheme

Scheme

Marks

[\(\mathrm{E}(X) =\)] \(-1 \times p + 0 + 1 \times p + 3 \times 0.3 + 7r\) or \(0.9 + 7r\) [\(= 1.95\)]

3.1a

[\(7r = 1.05\) so] \(\underline{\boldsymbol{r = 0.15}}\)

1.1b

[Sum of probs = 1] \(2p + 0.3 + 2r = 1\) (o.e.)

2.1

\(\underline{\boldsymbol{p = 0.2}}\)

A1ft

1.1b

[Let \(Y = \sqrt{X+1}\)]

\(y\)	0	1	\(\sqrt{2}\)	2	\(\sqrt{8}\) or \(2\sqrt{2}\)
\(\mathrm{P}(Y = y)\)	“0.2”	“0.15”	“0.2”	0.3	“0.15”

[\(\mathrm{E}(Y) =\)] \(0 + \text{‘}0.15\text{’} + \text{‘}0.2\text{’}\sqrt{2} + 2 \times 0.3 + \text{‘}0.15\text{’} \times \sqrt{8}\)

3.1a

\(= \underline{\mathbf{0.75 + 0.5}}\sqrt{2}\)

A1cao

1.1b

(6 marks)

Notes

1^st M1 for an attempt at an expression for \(\mathrm{E}(X)\) at least 3 correct non-zero products

1^st A1 for \(r = 0.15\) oe

2^nd M1 for using sum of probabilities = 1 to form an equation for \(p\) (ft their value or letter \(r\))

2^nd A1ft for \(p = 0.2\) oe or ft their \(r\) for \(p = \dfrac{0.7 - 2 \times \text{“}0.15\text{”}}{2}\) (provided \(p\) is a probability)

3^rd M1 for attempt at \(\mathrm{E}(Y)\) with at least 2 correct \(y\) values and products ft their \(p\) and \(r\)

3^rd A1 cao allow fraction or decimal, also allow \(\dfrac{3 + 2\sqrt{2}}{4}\)