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Cumulative distribution function

Cumulative distribution function

IB Year 5 | Grade 11

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

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

  1. Find the median of \(X\).
  2. Find \(\mathrm{f}(x)\), the probability density function of \(X\). Hence:
    1. Show that \(\mathrm{E}(X) = \frac{1}{3}\).
    2. Find \(\mathrm{Var}(X)\).
  3. Show that the median of \(\sqrt{X}\) and the mean of \(\sqrt{X}\) are equal.

Solution:

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Answer:(a) \(\frac14\). (b) \(\mathrm{f}_X(x)=\frac{1}{2\sqrt{x}}\) for \(0<x<1\), zero otherwise. (i) \(\mathrm{E}(X)=\frac13\). (ii) \(\mathrm{Var}(X)=\frac4{45}\). (c) Mean and median of \(\sqrt{X}\) both equal \(\frac12\).

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