S2 June 2005 Q2

2. The continuous random variable \(X\) is uniformly distributed over the interval \([2, 6]\).

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

Find

(b) \(\mathrm{E}(X)\), (1)

(d) the cumulative distribution function of \(X\), for all \(x\), (4)

(e) \(\mathrm{P}(2.3 \lt X \lt 3.4)\). (2)

Mark scheme (a)

Scheme	Marks
\(\mathrm{f}(x) = \dfrac{1}{4}, \quad 2 \leqslant x \leqslant 6\)	B1
\(\phantom{\mathrm{f}(x)} = 0, \quad \text{otherwise}\)	B1
	(2)

1st B1 \(\dfrac{1}{4}\) and range

2nd B1 0 and range

Scheme	Marks
\(\mathrm{E}(X) = 4\) by symmetry or formula	B1
	(1)

Mark scheme (d)

Scheme	Marks
\(\mathrm{F}(x) = \displaystyle\int_2^x \frac{1}{4}\,\mathrm{d}t = \left[\frac{1}{4}t\right]_2^x\)	M1
\(= \dfrac{1}{4}(x - 2)\)	A1
\(\mathrm{F}(x) = \dfrac{1}{4}(x - 2), \quad 2 \leqslant x \leqslant 6\)	B1ft
\(\phantom{\mathrm{F}(x)} = 1, \quad x \gt 6\) \(\phantom{\mathrm{F}(x)} = 0, \quad x \lt 2\)	B1
	(4)

M1 use of \(\int \mathrm{f}(x)\,\mathrm{d}x\)

A1 \(\frac{1}{4}(x - 2)\) or equiv.

B1ft \(\frac{1}{4}(x - 2)\) and range

B1 ends and ranges