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\includegraphics[max width=\textwidth, alt={}, center]{054e0081-afce-4a87-93f5-650dad40b313-3_508_611_262_719}
The diagram shows the probability density function f of the continuous random variable \(T\), given by
$$f ( t ) = \begin{cases} a t & 0 \leqslant t \leqslant 1
a & 1 < t \leqslant 4
0 & \text { otherwise } \end{cases}$$
where \(a\) is a constant.
- Find the value of \(a\).
- Obtain the cumulative distribution function of \(T\).
- Find the cumulative distribution of \(Y\), where \(Y = T ^ { \frac { 1 } { 2 } }\), and hence find the probability density function of \(Y\).