OCR S3 2006 June — Question 4

Exam BoardOCR
ModuleS3 (Statistics 3)
Year2006
SessionJune
TopicContinuous Probability Distributions and Random Variables
TypeFind median or percentiles

4 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 4 } { 3 x ^ { 3 } } & 1 \leqslant x < 2
\frac { 1 } { 12 } x & 2 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$
  1. Find the upper quartile of \(X\).
  2. Find the value of \(a\) for which \(\mathrm { E } \left( X ^ { 2 } \right) = a \mathrm { E } ( X )\).