I'm calculating the MSE of a particular statistic and I'm ultimately stuck on deriving the following: $$E\left[\sum_{i=1}^n(X_i-X_{n:1})^2\right]$$
More specifically, to calculate the expectation $E[g(x_i, x_{n ;1})],$ would I need to use the joint pdf for $X$ and $X_{n:1}$? I don't believe I know how to derive this. Also, I am assuming that the two arguments for the function $g$ are not independent, which may or may not be correct...