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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

\[\mathrm{f}(x) = \begin{cases} k(2 - x), & \text{for}\hspace{0.5em} 0 \le x \le 2 \\ 0, & \text{otherwise} \end{cases} , \text{ where}\hspace{0.5em} k \text{ is a constant.}\]

  1. Find, in terms of \(k\), the cumulative distribution function.
  2. Hence or otherwise, find the value of \(k\).

Solution:

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Answer:(a) \(F(x)=\begin{cases}0,&x<0,\\ k\left(2x-\dfrac{x^2}{2}\right),&0\le x\le2,\\ 2k,&x>2.\end{cases}\) (b) \(k=\dfrac12\).

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