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A real probability distribution is symmetric about $m$ when $X-m$ and $m-X$ have the same distribution. A centered normal distribution is symmetric about zero.

A truncated normal distribution is a normal distribution conditioned to lie in a specified subset of the real line. Its density is the original Gaussian density divided by the probability of the retained set.

For a normal distribution $N_n(\mu ,{\sigma }^{2}I)$ with unrestricted $\mu \in {\mathbb{R}}^n$ and unknown ${\sigma }^2$, an integrable data-only unbiased estimator of the mean-vector prediction risk of a fixed rank-$k$ orthogonal projection matrix exists exactly when $2k=n$. To prove necessity, randomize the mean by independent Gaussian variance $v$: conditioning adds $(n-k)v$ to the squared bias term, whereas the marginal variance identity would add $kv$. When $2k=n$, the residual sum of squares itself is unbiased.