$($b$)$ Show that for the MLE ${}_{\text{M}\text{L}\text{E}}$ of a parameter $in{R}^d$ and any known bijective function $g,$ the MLE of $g()$ is

$g({}_{\text{M}\text{L}\text{E}}).$ It turns out that this holds for any function $g,$ i$.$e$.g$ need not be bijective $($you do not need to show this

but you should try to understand why this is true$).$ Use this fact to find the MLE of $e^{}$ in the setting of part $($a$).$

$($c$)$ Sometimes we have prior knowledge concerning the value of the parameter $.$ This is often encoded as a

prior distribution characterized by $p().$ In such cases one usually computes the MAP $($maximum a posteriori$)$

estimate of $,$ defined as ${}_{\text{M}\text{A}\text{P}}=argma{x}_{}p(|x_1,x_2,$dots,$x_n),$ instead of the MLE.

In the setting of part $($a$)$ suppose that we know from prior information that $$ is likely small. Specifically, we

model this with the prior $p()=2e^{-{}^2}{}^{-\frac{1}{2}}.$ Compute the MAP estimate of $.$

Hint: First show that ${}_{\text{M}\text{A}\text{P}}=argma{x}_{}p(x_1,x_2,$dots,$x_n|)p().$