S2 June 2009 Q6

EdexcelOld spec13 marksContinuous Random Variables

6. The three independent random variables \(A\), \(B\) and \(C\) each has a continuous uniform distribution over the interval [0, 5].

(a) Find \(\mathrm{P}(A \gt 3)\). (1)

(b) Find the probability that \(A\), \(B\) and \(C\) are all greater than 3. (2)

The random variable \(Y\) represents the maximum value of \(A\), \(B\) and \(C\).

The cumulative distribution function of \(Y\) is

\[\mathrm{F}(y) = \begin{cases} 0 & y \lt 0 \\[1mm] \dfrac{y^3}{125} & 0 \leqslant y \leqslant 5 \\[2mm] 1 & y \gt 5 \end{cases}\]

(d) Sketch the probability density function of \(Y\). (2)

(e) Write down the mode of \(Y\). (1)

(f) Find \(\mathrm{E}(Y)\). (3)

(g) Find \(\mathrm{P}(Y \gt 3)\). (2)

Scheme	Marks
\(\mathrm{P}(A \gt 3) = \tfrac{2}{5} = 0.4\)	B1
	(1)

Scheme	Marks
\((0.4)^3\ ,= 0.064\) or \(\dfrac{8}{125}\)	M1, A1
	(2)

Mark scheme (c)

Scheme	Marks
\(\mathrm{f}(y) = \dfrac{\mathrm{d}}{\mathrm{d}y}(\mathrm{F}(y) = \begin{cases} \dfrac{3y^2}{125} & 0 \leqslant y \leqslant 5 \\[2mm] 0 & \textit{otherwise} \end{cases}\)	M1A1
	(2)

Notes

M1 for attempt to differentiate the cdf. They must decrease the power by 1
A1 fully correct answer including 0 otherwise. Condone < signs

Sketch: f(y) = 0 for y below 0, a curve from (0, 0) increasing and curving upwards to y = 5, then 0 for y above 5 — Mark scheme (d)

Scheme	Marks
	B1 B1
	(2)

Scheme	Marks
Mode = 5	B1
	(1)

Mark scheme (f)

Scheme	Marks
\(\mathrm{E}(Y) = \displaystyle\int_0^5 \left(\dfrac{3y^3}{125}\right)\mathrm{d}y = \left[\dfrac{3y^4}{500}\right]_0^5 = \dfrac{15}{4}\) or 3.75	M1M1A1
	(3)

Notes

1^st M1 for attempt to integrate their \(y\mathrm{f}(y)\) \(y^n \to y^{n+1}\).
2^nd M1 for attempt to use correct limits
A1 cao

Mark scheme (g)

Scheme	Marks
\(\mathrm{P}(Y \gt 3) = \begin{cases} \displaystyle\int_3^5 \tfrac{3y^2}{125}\,\mathrm{d}y \\ \text{or } 1 - \mathrm{F}(3) \end{cases} = 1 - \dfrac{27}{125} = \dfrac{98}{125} = 0.784\)	M1A1
	(2)
	(13 marks)

Notes

M1 for attempt to find \(\mathrm{P}(Y \gt 3)\).
e.g. writing \(\displaystyle\int_3^5 \textit{their } f(y)\) must have correct limits
or writing \(1 - \mathrm{F}(3)\)