Let $X_1,\ldots, X_n$ be IID random variables with mean 0 and variance 1, and $X_1',\ldots, X_n'$ an independent copy of this sequence. Define $$W=\frac{1}{\sqrt n}\sum_{i=1}^nX_i$$ and $$W'=W-\frac{1}{\sqrt n}X_I+ \frac{1}{\sqrt n}X_I'$$ where $I$ is randomly chosen from $\lbrace 1,\ldots ,n \rbrace$ and $X_I$ is independent and equal in distribution to other $X_i$s. Let $F=\sigma(X_1,\ldots ,X_n)$. How can I verify
$$\mathbb{E}[\frac{n}{2}(W'-W)^2\vert F ]= \frac{1}{2}+\frac{1}{2n}\sum_{i=1}^nX_i^2 \,?$$
I have $\frac{n}{2}(W'-W)^2=\frac{1}{2}(X_I'-X_I)^2$, but I don't know how to compute the conditional expectation $\mathbb{E}[\frac{1}{2}(X_I'-X_I)^2\vert F].$