(Solved): PLEASE SHOW CLEAR GOOD WORK! THANK YOU Optimality of the mean. One fact that we used implicitly i ...

PLEASE SHOW CLEAR GOOD WORK! THANK YOU

Optimality of the mean. One fact that we used implicitly in the lecture is the following: If we want to summarize a bunch of numbers $$x_1,\dots ,x_n$$ by a single number $$s$$, the best choice for $$s$$, the one that minimizes the average squared error, is the mean of the $$x_i$$ 's. Let's see why this is true. We begin by defining a suitable loss function. Any value $$s?\mathbb{R}$$ induces a mean squared loss (MSE) given by: $$L(s)=\frac{1}{n}\underset{i=1}{\overset{n}{?}}{\left(x_i?s\right)}^2.$$ We want to find the $$s$$ that minimizes this function. (a) Compute the derivative of $$L(s)$$. (b) What value of $$s$$ is obtained by setting the derivative $$dL/ds$$ to zero?