CAIE FP2 2010 June — Question 7

Exam BoardCAIE
ModuleFP2 (Further Pure Mathematics 2)
Year2010
SessionJune
TopicCumulative distribution functions
TypeCalculate probabilities from CDF

7 The continuous random variable \(X\) has distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 0
1 - \mathrm { e } ^ { - \frac { 1 } { 2 } x } & x \geqslant 0 \end{cases}$$ For a random value of \(X\), find the probability that 2 lies between \(X\) and \(4 X\). Find also the expected value of the width of the interval ( \(X , 4 X\) ).