<p>The continuous random variable \(X\) has cumulative distribution function given by</p>
<p>\[F(x) = \begin{cases} 0 & , & x \le 1 \\ k(x - 1)^3 & , & 1 &lt; x &lt; 3 \\ 1 & , & x \ge 3 \end{cases}\]</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the value of \(k\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find in a form not involving \(k\), the lower and upper quartile of \(X\), giving your answers correct to \(3\) significant figures.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the probability density function of \(X\).</span></div></li><li><div class="list-item-paragraph">Find \(\mathrm{E}(X - 1)\), and hence write down the value of \(\mathrm{E}(X)\).</div></li></ol>

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

\[F(x) = \begin{cases} 0 & , & x \le 1 \\ k(x - 1)^3 & , & 1 < x < 3 \\ 1 & , & x \ge 3 \end{cases}\]

Find in a form not involving \(k\), the lower and upper quartile of \(X\), giving your answers correct to \(3\) significant figures.

Find the probability density function of \(X\).

Find \(\mathrm{E}(X - 1)\), and hence write down the value of \(\mathrm{E}(X)\).

Calculating median and IQR

Calculating median and IQR