Assume that $Y, Z$ are independent, zero-mean, continuous random variables with variances $\sigma_{Y}^{2}$ and $\sigma_{Z}^{2},$ respectively. Let $X=Y+Z$. The optimal value of $\alpha$ which minimizes $\mathbb{E}\left[(X-\alpha Y)^{2}\right]$ is
- $\frac{\sigma_{Y}^{2}}{\sigma_{Y}^{2}+\sigma_{Z}^{2}}$
- $\frac{\sigma_{Z}^{2}}{\sigma_{Y}^{2}+\sigma_{Z}^{2}}$
- $1$
- $\frac{\sigma_{Y}^{2}}{\sigma_{Z}^{2}}$
- None of the above.