$2.$e Consider the example problem given as a solution to Exercise $2.$c above

$($this solution is provided separately$).$ Determine the marginal probability

density for $y({}_{\text{n}\text{e}\text{w}})$ when using algorithm A and verify analytically that this

density integrates to unity on the relevant domain. In addition, construct a

plot of the marginal density for $y({}_{\text{n}\text{e}\text{w}})$ and comment on why the form of

this density shows that algorithm A does not converge to ${}^{**}.($Note: the

density is not continuous at $y=0.)$

$2.$c Consider the standard random search algorithms in Section $2\mathrm{.}2$ of ISSO with

the exception of using noisy loss measurements $y()$ in place of $L()$ values.

Assume that $()$ has mean zero.

$($a$)$ Describe an example problem $($i$.$e$.,$ describe $,L(),$ and $())$ where

such a noisy implementation of random search leads to an algorithm

that is guaranteed to not converge to ${}^{**}$ with probability $>0.$ The

rationale for nonconvergence may be conceptual, diagrammatic, or

mathematical, as appropriate. The ideas should apply to all of

algorithms A$,$ B$,$ and C$.$

$($b$)$ For blind random search $($algorithm A$)$ with uniformly distributed hat$({)}_0$

and candidate $,$ provide exact values or bounds to relevant probabilities

that show tha

t the algorithm does not converge to ${}^{**}$ in the problem of

part $($a$).$

$2.$e PLEASE!