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Calculating median and IQR

Calculating median and IQR

IB Year 5 | Grade 11

The continuous random variable \(X\) has cumulative distribution function given by

\[F(x) = \begin{cases} 0 & , & x \le 1 \\ k(x - 1)^3 & , & 1 < x < 3 \\ 1 & , & x \ge 3 \end{cases}\]

  1. Find the value of \(k\).
  2. Find in a form not involving \(k\), the lower and upper quartile of \(X\), giving your answers correct to \(3\) significant figures.
  3. Find the probability density function of \(X\).
  4. Find \(\mathrm{E}(X - 1)\), and hence write down the value of \(\mathrm{E}(X)\).

Solution:

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Answer:(a) \(k=\dfrac18\). (b) Lower quartile \(2.26\), upper quartile \(2.82\) (3 s.f.). (c) \(\mathrm f(x)=\begin{cases}\dfrac38(x-1)^2,&1<x<3,\\ 0,&\text{otherwise}.\end{cases}\) (d) \(\mathrm E(X-1)=\dfrac32,\quad\mathrm E(X)=\dfrac52\).

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