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Calculating median and IQR of a CRV

Calculating median and IQR of a CRV

IB Year 5 | Grade 11

If \(X\) is a continuous random variable with probability density function \(\mathrm{f}(x) = \frac{1}{8}x\), where \(0 \leq x \leq 4\).

  1. Find the cumulative distribution function \(F(x)\) and sketch \(y = F(x)\).
  2. Find the median \(m\), the upper quartile \(s\) and the lower quartile \(r\).
  3. Find \(\mathrm{P}(0.3 < X \leq 1.8)\)

Solution:

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Answer:(a) \(\mathrm{F}(x)=\begin{cases}0,&x<0,\\ \frac{x^2}{16},&0\leq x\leq4,\\ 1,&x>4.\end{cases}\) Sketch in the working. (b) \(m=2\sqrt2\), \(s=2\sqrt3\), \(r=2\). (c) \(\frac{63}{320}\approx0.197\) (3 s.f.).

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