The continuous random variable \(X\) has probability density function f given by
\[f(x)=\left\{\begin{array}{ll}
\frac{3}{4 x^{2}}+\frac{1}{4} & 1 \leqslant x \leqslant 3 \\
0 & \text { otherwise }
\end{array}\right.\]
(i) Find the distribution function of \(X\).
(ii) Find the exact value of the interquartile range of \(X\).

The random variable \(X\) has probability density function f given by
\[\mathrm{f}(x)=\left\{\begin{array}{ll}
0.2 \mathrm{e}^{-0.2 x} & x \geqslant 0 \\
0 & \text { otherwise }
\end{array}\right.\]
(i) Find the distribution function of \(X\).

(ii) Find \(\mathrm{P}(X>2)\).

(iii) Find the median of \(X\).

The continuous random variable \(X\) has distribution function given by
\[\mathrm{F}(x)=\left\{\begin{array}{ll}
1-\mathrm{e}^{-0.4 x} & x \geqslant 0 \\
0 & \text { otherwise }
\end{array}\right.\]
(i) Find \(\mathrm{P}(X>2)\).
(ii) Find the interquartile range of \(X\).

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

\(F(x)=\begin{cases}0, & x\lt 1,\\ \frac{1}{5}x+a, & 1\leqslant x\lt 4,\\ \frac{1}{50}x^2+b, & 4\leqslant x\leqslant 6,\\ 1, & x\gt 6,\end{cases}\)

where \(a\) and \(b\) are constants.

(a) Find the value of \(a\) and the value of \(b\).

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

(c) Given that \(\mathrm{E}(X)=\frac{529}{150}\), find \(\operatorname{Var}(X)\).

(d) Find the 10th and 90th percentiles of \(X\).

The continuous random variable \(X\) has cumulative distribution function F given by
\(F(x)=\left\{\begin{array}{ll} 0 & x\lt 2, \\ \frac{(x-2)^{2}}{12} & 2 \leqslant x\lt 4, \\ 1-\frac{(8-x)^{2}}{24} & 4 \leqslant x \leqslant 8, \\ 1 & x\gt 8 . \end{array}\right.\)
(a) Sketch the graph of the probability density function of \(X\).

(b) Find \(\mathrm{E}(X)\).

(c) Find the exact value of the interquartile range of \(X\).

The continuous random variable \(X\) has probability density function \(f\) given by
\(f(x)=\left\{\begin{array}{ll} \frac{1}{128}\left(4ax-bx^{3}\right) & 0 \leqslant x \leqslant 4 \\ c & 4 \leqslant x \leqslant 6 \\ 0 & \text{otherwise} \end{array}\right.\)
where \(a\), \(b\) and \(c\) are constants.

The upper quartile of \(X\) is equal to 4.

(a) Show that \(c=\frac{1}{8}\) and find the values of \(a\) and \(b\).

(b) Find the exact value of the median of \(X\).

(c) Find \(\mathrm{E}(\sqrt{X})\), giving your answer correct to 2 decimal places.

The continuous random variable \(X\) has probability density function \(f\) given by
\(\mathrm{f}(x)=\left\{\begin{array}{ll} k x(4-x) & 0 \leqslant x\lt 2 \\ k(6-x) & 2 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{array}\right.\)
where \(k\) is a constant.
(a) Show that \(k=\frac{3}{40}\).

(b) Given that \(\mathrm{E}(X)=2.5\), find \(\operatorname{Var}(X)\).

(c) Find the median value of \(X\).

The continuous random variable \(X\) has cumulative distribution function F given by
\(F(x)=\left\{\begin{array}{ll} 0 & x\lt 0, \\ \frac{1}{81} x^{2} & 0 \leqslant x \leqslant 9, \\ 1 & x\gt 9 . \end{array}\right.\)
(a) Find \(\mathrm{E}(\sqrt{X})\).

(b) Find \(\operatorname{Var}(\sqrt{X})\).

(c) The random variable \(Y\) is given by \(Y^{3}=X\). Find the probability density function of \(Y\).

The continuous random variable \(X\) has cumulative distribution function F given by
\(F(x)=\left\{\begin{array}{lc} 0 & x\lt -1, \\ \frac{1}{2}(1+x)^{2} & -1 \leqslant x \leqslant 0, \\ 1-\frac{1}{2}(1-x)^{2} & 0\lt x \leqslant 1, \\ 1 & x\gt 1 . \end{array}\right.\)
(a) Find the probability density function of \(X\).

(b) Find \(\mathrm{P}\left(-\frac{1}{2} \leqslant X \leqslant \frac{1}{2}\right)\).

(c) Find \(\mathrm{E}\left(X^{2}\right)\).

(d) Find \(\operatorname{Var}\left(X^{2}\right)\).

The continuous random variable \(X\) has probability density function \(f\) given by
\(f(x)=\left\{\begin{array}{ll} a+\frac{1}{5} x & 0 \leqslant x\lt 1 \\ 2 a-\frac{1}{5} x & 1 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{array}\right.\)
where \(a\) is a constant.
(a) Find the value of \(a\).

(b) Find \(\mathrm{E}\left(X^{2}\right)\).

(c) Find the cumulative distribution function of \(X\).

The continuous random variable \(X\) has probability density function \(f\) given by
\(f(x)=\left\{\begin{array}{ll} \frac{1}{5} x & 0 \leqslant x\lt 2 \\ \frac{2}{15}(5-x) & 2 \leqslant x \leqslant 5 \\ 0 & \text { otherwise } \end{array}\right.\)
(a) Find the cumulative distribution function of \(X\).

(b) Find the median value of \(X\).

(c) Find \(\mathrm{E}\left(X^{2}\right)\).

(d) Find \(\mathrm{P}(1 \leqslant X \leqslant 3)\).

The continuous random variable \(X\) has probability density function f given by
\(f(x)=\left\{\begin{array}{ll} k x^{3} & 0 \leqslant x\lt 1 \\ k(5-x) & 1 \leqslant x \leqslant 5 \\ 0 & \text { otherwise } \end{array}\right.\)
where \(k\) is a constant.
(a) Sketch the graph of f.

(b) Show that \(k=\frac{4}{33}\).

(c) Find the cumulative distribution function of \(X\).

(d) Find the median value of \(X\).

The continuous random variable \(X\) has probability density function f given by
\(\mathrm{f}(x)=\left\{\begin{array}{ll} k x & 0 \leqslant x\lt 1, \\ k x^{2} & 1 \leqslant x \leqslant 2, \\ 0 & \text { otherwise. } \end{array}\right.\)
(a) Show that \(k=\frac{6}{17}\).

(b) Find the cumulative distribution function of \(X\).

(c) Find the median value of \(X\).

(d) Find \(\mathrm{E}\left(\frac{1}{X}\right)\).