10 The continuous random variable \( X \) has the cumulative distribution function
\[
F(x)=\left\{\begin{array}{ll}
0, & x<0 \\
a \sqrt{x^{3}}-3 x^{2}, & 0 \leq x \leq 1 \\
1, & x>1
\end{array}\right.
\]
where \( a \) is constant. Show that \( a=4 \).
\( [2 \) marks \( ] \)
Hence,
(a) calculate the mean and variance of \( X \).
\( [6 \) marks \( ] \)
(b) find \( P\left[X-E(X)<\frac{1}{10}\right] \).
\( [2 \) marks \( ] \)
(c) if \( Y=4 X-3 \), find the \( \mathrm{E}(Y) \) and \( \operatorname{Var}(Y) \).
[5 marks]
