Further Mathematics 7367: Continuous Random Variables Topical Past Paper Questions

The continuous random variable X has probability density function $$ \mathrm{f}(x)=\left\{\begin{aligned}&\frac{1}{6}\mathrm{e}^{\frac{x}{3}}&\quad&0\leq x\leq\ln27\\ &\\ &0&\quad&otherwise\end{aligned}\right. $$ Show that the mean of $X$ is $\frac{3}{2}(\ln 27 - 2)$

The continuous random variable X has probability density function $$ \mathrm{f}(x)=\left\{\begin{aligned}&k\sin2x&\quad&0\leq x\leq\frac{\pi}{6}\\ &0&\quad&otherwise\end{aligned}\right. $$ where k is a constant. (a) Show that k = 4 [4 marks] Find the cumulative distribution funct…

The continuous random variable $X$ has cumulative distribution function $F(x)$ where $$ \mathrm{F}(x)=\left\{\begin{aligned}&0&&x=0\\ &\mathrm{e}^{kx}-1&&0\leq x\leq5\\ &1&&x>5\end{aligned}\right. $$ Show that $ k = \frac{1}{5} \ln 2 $ [2 marks] (b) Show that the median of X is $…