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Cumulative distribution function

Cumulative distribution function

IB Year 5 | Grade 11

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

\[\mathrm{f}(x) = \begin{cases} 4k, & 0 \le x \le 1 \\ k(x - 3)^2, & 1 \le x \le 3 \\ 0, & \text{otherwise} \end{cases}\]

  1. Find the value of \(k\) and hence show that the median of \(X\) is \(\frac{5}{6}\).
  2. Show that \(\mathrm{E}(X) = 0.9\) and deduce the value of \(c\), given that \(\mathrm{E}(X + c) = 3\mathrm{E}(X - c)\).
  3. Given that \(x_1\) and \(x_2\) are two independent observations of \(X\), find the probability that
    1. both are greater than \(1\),
    2. one is less than \(0.5\) and the other is greater than \(2\).
  4. Find the cumulative distribution function of \(X\).

Solution:

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Answer:(a) \(k=\frac3{20}\), median \(\frac56\) (shown). (b) \(\mathrm{E}(X)=0.9\) (shown), \(c=\frac9{20}\). (c)(i) \(\frac4{25}\). (c)(ii) \(\frac3{100}\). (d) \(\mathrm{F}(x)=\begin{cases}0,&x<0,\\ \frac{3x}{5},&0\leq x\leq1,\\ 1+\frac{(x-3)^3}{20},&1<x<3,\\ 1,&x\geq3.\end{cases}\)

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