S1 January 2007 Q3

EdexcelOld spec13 marksDRVs

3. The random variable \(X\) has probability function

\[\mathrm{P}(X = x) = \frac{(2x - 1)}{36} \qquad x = 1, 2, 3, 4, 5, 6.\]

(a) Construct a table giving the probability distribution of \(X\). (3)

Find

(b) \(\mathrm{P}(2 \lt X \leqslant 5)\), (2)

(d) Show that \(\mathrm{Var}(X) = 1.97\) to 3 significant figures. (4)

(e) Find \(\mathrm{Var}(2 - 3X)\). (2)

Mark scheme (a)

Scheme

Marks

\(x\)	1	2	3	4	5	6
\(\mathrm{P}(X = x)\)	\(\dfrac{1}{36}\)	\(\dfrac{3}{36}\)	\(\dfrac{5}{36}\)	\(\dfrac{7}{36}\)	\(\dfrac{9}{36}\)	\(\dfrac{11}{36}\)

0.0278, 0.0833, 0.139, 0.194, 0.25, 0.306 (Accept awrt 3 s.f)

B1, B1, B1

(3)

Notes

N.B. Part (a) doesn’t have to be in a table, could be a list \(\mathrm{P}(X = 1) = \ldots\)etc

1^st B1 for \(x = 1, \ldots 6\) and at least one correct probability N.B. \(\tfrac{3}{36} = \tfrac{1}{12}\) and \(\tfrac{9}{36} = \tfrac{1}{4}\)

2^nd B1 for at least 3 correct probabilities

3^rd B1 for a fully correct probability distribution.

Mark scheme (b)

Scheme	Marks
\(\mathrm{P}(3) + \mathrm{P}(4) + \mathrm{P}(5) =, \ \underline{\dfrac{21}{36} \text{ or } \dfrac{7}{12}}\) or awrt 0.583	M1, A1
	(2)

Notes

M1 for attempt to add the correct three probabilities, ft their probability distribution

Mark scheme (c)

Scheme	Marks
\(\mathrm{E}(X) = \dfrac{1}{36} + 2 \times \dfrac{3}{36} + \ldots, = \dfrac{161}{36}\) or \(4.47\dot{2}\) or \(4\tfrac{17}{36}\)	M1, A1
	(2)

Notes

M1 for a correct attempt at \(\mathrm{E}(X)\). Minimum is as printed. Exact answer only scores M1A1.
[Division by 6 at any point scores M0, no ISW. Non-exact answers with no working score M0.]

Mark scheme (d)

Scheme	Marks
\(\mathrm{E}(X^2) = \dfrac{1}{36} + 2^2 \times \dfrac{3}{36} + \ldots, = \dfrac{791}{36}\) or full expression or \(21\tfrac{35}{36}\) or awrt 21.97	M1, A1
\(\mathrm{Var}(X) = \dfrac{791}{36} - \left(\dfrac{161}{36}\right)^2, \ = \underline{\mathbf{1.9714\ldots}}\) *	M1, A1c.s.o.
	(4)

Notes

1^st M1 for a correct attempt at \(\mathrm{E}(X^2)\). Minimum as printed. \(\dfrac{791}{36}\) or awrt 21.97 scores M1A1.

2^nd M1 for their \(\mathrm{E}(X^2) - (\text{their } \mathrm{E}(X))^2\).

2^nd A1 cso needs awrt 1.97 and \(\dfrac{791}{36} - \left(\dfrac{161}{36}\right)^2\) or \(\dfrac{2555}{1296}\) or any fully correct expression seen.

Can accept at least 4 sf for both. i.e. 21.97 for \(\dfrac{791}{36}\), 4.472 for \(\dfrac{161}{36}\), 20.00 for \(\left(\dfrac{161}{36}\right)^2\).

Mark scheme (e)

Scheme	Marks
\(\mathrm{Var}(2 - 3X) = 9 \times 1.97\) or \((-3)^2 \times 1.97, \ = 17.73\)	M1, A1
	(2)
	(13 marks)

Notes

A1 awrt 17.7 or \(\tfrac{2555}{144}\)

M1 for correct use of \(\mathrm{Var}(aX + b)\) formula or a full method.

NB \(-3^2 \times 1.97\) followed by awrt 17.7 scores M1A1 BUT \(-3^2 \times 1.97\) alone, or followed by −17.7, scores M0A0.