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**lllll** I'm having trouble solving the following problem correctly:

The waiting time $\displaystyle Y$ until delivery of a new component for an industrial operation is uniformly distributed over the interval 1 to 5 days. The cost of this delay is given by $\displaystyle U = 2Y^2+3$. Find the probability density function for $\displaystyle U$.

so far I have:

$\displaystyle Y= \pm \sqrt{\frac{u-3}{2}}$

and based on an example from the book I have:

$\displaystyle f_U(u) = \left\{ \begin{array}{rcl}

\frac{1}{2\sqrt{u}} \left[f_Y\left(\sqrt{u}\right) + f_Y(y)(-\sqrt{u})\right] & \mbox{for} & u>0 \\

0 & \mbox{if} & \mbox{other}

\end{array}\right.$

then combining for the top part I would get:

$\displaystyle =\frac{1}{2\sqrt{\frac{u-3}{2}}} \left[\frac{1}{4}\left(\sqrt{\frac{u-3}{2}}\right) + \frac{1}{4}\left(-\sqrt{\frac{u-3}{2}}\right)\right]$

$\displaystyle =\frac{1}{8\sqrt{\frac{u-3}{2}}}$

and based on the solution in the back of the book it's:

$\displaystyle =\frac{1}{{\color{red}16}\sqrt{\frac{u-3}{2}}}$

if anyone can help me it it would be greatly appreciated