Suppose $$ is a random variable and that given $=,$ the real observa$-$

tion $Y$ has a probability density function $($pdf$)p(\frac{y}{})=\frac{}{2}*exp(-|y|),$yin

$R.$

Given that $$ has a prior pdf: $p()=\{\begin{array}{cc}\frac{1}{},& 1e\\ 0& \text{o}\text{t}\text{h}\text{e}\text{r}\text{w}\text{i}\text{s}\text{e}& \end{array}$

$($i$)$ Find the maximum a posteriori estimate of $$ given $Y.$

$($ii$)$ Find the minimum mean square error estimate of $$ given $Y.$

Hint: ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}$xexp$(cx)dx=$exp$(c*x)(\frac{c*x-1}{c^2})$