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Cumulative distribution function

Cumulative distribution function

IB Year 5 | Grade 11

The continuous random variable \(X\) has probability density function given by

\[\mathrm{f}(x) = \begin{cases} \frac{\pi}{20} \cos\left( \frac{\pi x}{20} \right), & \text{for}\hspace{0.5em} 0 \le x \le 10 \\ 0, & \text{otherwise} \end{cases}\]

  1. Find the cumulative distribution \(F(x)\) and sketch \(y = F(x)\).
  2. Find the median \(m\).

Solution:

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Answer:(a) \(\mathrm{F}(x)=\begin{cases}0,&x<0,\\ \sin\left(\frac{\pi x}{20}\right),&0\leq x\leq10,\\ 1,&x>10.\end{cases}\) Sketch in the working. (b) \(m=\frac{10}{3}\).

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