Suppose $Y=X+Z$, where $X$ and $Z$ are independent zero-mean random variables each with variance $1.$ Let $\hat{X}(Y)=a Y$ be the optimal linear least-squares estimate of $X$ from $Y$, i.e., $a$ is chosen such that $E\left[(X-a Y)^{2}\right]$ is minimized. What is the resulting minimum $E\left[(X-\hat{X}(Y))^{2}\right]?$
- $1$
- $\frac{2}{3}$
- $\frac{1}{2}$
- $\frac{1}{3}$
- $\frac{1}{4}$