CAIE S2 2024 June — Question 5

Exam BoardCAIE
ModuleS2 (Statistics 2)
Year2024
SessionJune
TopicContinuous Probability Distributions and Random Variables
TypeSingle-piece PDF with k
5 A random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} a x - x ^ { 3 } & 0 \leqslant x \leqslant \sqrt { 2 }
0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.
  1. Show that \(a = 2\) .
  2. Find the median of \(X\) .
  3. Find the exact value of \(\mathrm { E } ( X )\).
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