Each of the random variables $T$, $U$, $V$, $W$, $X$, $Y$ and $Z$ takes values between 0 and 1 only. Their probability density functions are shown in Figs 1 to 7 respectively.

a)(i) Which of these variables has the largest median?

(ii) Which of these variables has the largest standard deviation? Explain your answer.

b) Use Fig. 2 to find $P\left( {U \lt 0.5} \right)$.

c) The probability density function of $X$ is given by

$f\left( x \right) = \left\{ \begin{gathered}
  \alpha {x^n}\,\,\,\,\,\,\,0 \leqslant x \leqslant 1, \hfill \\
  0\,\,\,\,\,\,\,\,\,\,\,\,\,\,otherwise, \hfill \\ 
\end{gathered}  \right.$

where $\alpha $ and $n$ are positive constants.

(i) Show that $\alpha  = n + 1$.

(ii) Given that $E\left( X \right) = \frac{5}{6}$, find $\alpha $ and $n$.