Problem $1[20$ points$]$ Linear Regression I

We would like to fit a linear regression estimate to the dataset with $x$ and $Y$ two variables, and

$Y=wx$ where $w$ is a constant showing the weight. Given $n$ independent sample pairs, $(x^{(1)},y^{(1)}),$

$(x^{(2)},y^{(2)}),$dots,$(x^{(n)},y^{(n)}),$ we aim to estimate $w$ by solving the following problem.

$w^{**}=argmi{n}_w\frac{1}{2}({\displaystyle \underset{i=1}{\overset{n}{}}}(y^{(i)}-w{x}^{(i)}{)}^2+w^2)$

where $0$ is a tuning hyperparameter.

a$)$ Give a solution in explicit formula for $w^{**}$ so that $w^{**}$ can be computed using only the training

data and $.$ Hint: take the derivative of the objective function and set it to zero.

b$)$ When $$ goes from $0$ to infinity, how does $w^{**}$ change? Give a brief explanation of your answer.