S2 January 2009 Q2

2. The continuous random variable \(X\) is uniformly distributed over the interval \([-2, 7]\).

(a) Write down fully the probability density function \(\mathrm{f}(x)\) of \(X\). (2)

(b) Sketch the probability density function \(\mathrm{f}(x)\) of \(X\). (2)

Find

(d) \(\mathrm{P}(-0.2 \lt X \lt 0.6)\). (2)

Scheme	Marks
\(\mathrm{f}(x) = \begin{cases} \frac{1}{9} & -2 \leqslant x \leqslant 7 \\ 0 & \textit{otherwise} \end{cases}\)	B1 B1
	(2)

Scheme	Marks
	B1 B1
	(2)

Mark scheme (c)

Scheme	Marks
\(\mathrm{E}(X) = \underline{2.5}\) \(\mathrm{Var}(X) = \tfrac{1}{12}(7 + 2)^2\) or \(\underline{6.75}\) both	B1
\(\mathrm{E}(X^2) = \mathrm{Var}(X) + \mathrm{E}(X)^2\)	M1
\(= 6.75 + 2.5^2\) \(= 13\)	A1
	(3)

Scheme	Marks
\(\displaystyle\int_{-2}^{7} x^2 f(x)\,dx = \left[\dfrac{x^3}{27}\right]_{-2}^{7}\)	B1 M1
\(= 13\)	A1

“\(\displaystyle\int x^2 f(x)\)” attempt to integrate and use limits of \(-2\) and 7