S2 June 2008 Q1

EdexcelOld spec10 marksContinuous Random Variables

1. Jean regularly takes a break from work to go to the post office. The amount of time Jean waits in the queue to be served at the post office has a continuous uniform distribution between 0 and 10 minutes.

(a) Find the mean and variance of the time Jean spends in the post office queue. (3)

(b) Find the probability that Jean does not have to wait more than 2 minutes. (2)

Jean visits the post office 5 times.

Jean is in the queue when she receives a message that she must return to work for an urgent meeting. She can only wait in the queue for a further 3 minutes.

Given that Jean has already been queuing for 5 minutes,

(d) find the probability that she must leave the post office queue without being served. (3)

Mark scheme (a)

Scheme	Marks
\(\mathrm{E}(X) = 5\)	B1
\(\mathrm{Var}(X) = \dfrac{1}{12}(10 - 0)^2\) or attempt to use \(\displaystyle\int \dfrac{x^2}{10}\,dx - \mu^2\)	M1
\(= \dfrac{100}{12} = \dfrac{25}{3} = 8\dfrac{1}{3} = 8.\dot{3}\) awrt 8.33	A1
	(3)

Notes

B1 cao

M1 using the correct formula \(\dfrac{(a - b)^2}{12}\) and subst in 10 or 0
or for an attempt at the integration they must increase the power of \(x\) by 1 and subtract their \(\mathrm{E}(X)\) squared.

A1 cao

Mark scheme (b)

Scheme	Marks
\(\mathrm{P}(X \leqslant 2) = (2 - 0) \times \dfrac{1}{10} = \dfrac{1}{5}\) or \(\dfrac{2}{10}\) or 0.2	M1 A1
	(2)

Notes

M1 for \(\mathrm{P}(X \leqslant 2)\) or \(\mathrm{P}(X \lt 2)\)
A1 cao

Mark scheme (c)

Scheme	Marks
\(\left(\dfrac{1}{5}\right)^5 = 0.00032\) or \(\dfrac{1}{3125}\) or \(3.2 \times 10^{-4}\) o.e.	M1 A1
	(2)

Notes

M1 (their b)⁵. If the answer is incorrect we must see this. No need to check with your calculator
A1 cao

Mark scheme (d)

Scheme	Marks
\(\mathrm{P}(X \geqslant 8)\) or \(\mathrm{P}(X \gt 8)\)	M1
\(\mathrm{P}(X \geqslant 8 \mid X \geqslant 5) = \dfrac{\mathrm{P}(X \geqslant 8)}{\mathrm{P}(X \geqslant 5)}\) \(= \dfrac{2/10}{5/10}\)	M1
\(= \dfrac{2}{5}\)	A1
	(3)
	(10 marks)

Notes

M1 writing \(\mathrm{P}(X \geqslant 8)\) (may use > sign). If they do not write \(\mathrm{P}(X \geqslant 8)\) then it must be clear from their working that they are finding it. 0.2 on its own with no working gets M0

M1 For attempting to use a correct conditional probability.

A1 2/5

Full marks for 2/5 on its own with no incorrect working

alternative

Scheme	Marks
remaining time \(\sim \mathrm{U}[0,5]\) or \(\mathrm{U}[5,10]\) \(\mathrm{P}(X \geqslant 3 \text{ or } 8) = \dfrac{2}{5}\)	M1 M1 A1

M1 for \(\mathrm{P}(X \geqslant 3)\) or \(\mathrm{P}(X \geqslant 8)\) may use > sign
M1 using either U[0,5] or U[5,10]
A1 2/5