Question

Show that $E\left[{(X-a)}^2\right]$ is minimized at $a=E\left[X\right]$.

Short answer

Differentiate$E\left[{(X-a)}^2\right]$respective to$a$.

Step-by-step solution

Given Information

$E\left[{(X-a)}^2\right]$is minimizing at$a=E\left[X\right]$.

Explanation

We have that

$E\left((X-a{)}^2\right)=E\left(X^2+a^2-2Xa\right)$

$=E\left(X^2\right)+a^2-2aE\left(X\right)$

Using the differentiation respective to $a$ and equalizing it to zero, we have that

$\frac{d}{da}E\left((X-a{)}^2\right)=2a-2E\left(X\right)$

$=0$

Explanation

which yields that $a=E\left(X\right)$.

So, $E\left((X-a{)}^2\right)$is minimized when$a=E\left(X\right)$ which had to be proved.

Final Answer

$E\left((X-a{)}^2\right)$ is minimized when$a=E\left(X\right)$ which had to be proved.