Explain conditional distribution given sufficient statistics. I understood proof of sufficient statistics by factorization theorem.
But I can't solve fllowing two questions about conditional distribution.


*

*When $X_1, X_2, ..., X_n \sim N(0, \sigma^2)$ are i.i.d., expain conditional distribution of $X=(X_1, X_2, ..., X_n)$ given $T=\sum_{i=1}^{n}X_i^2$.

*When $X_1, X_2, ..., X_n \sim N(\mu, \sigma^2)$ are i.i.d., expain conditional distribution of $X=(X_1, X_2, ..., X_n)$ given $T=(\bar{X}, \sum_{i=1}^{n}(X_i-\bar{X})^2)$.
I was given a hint of Q1, "Consider uniformaly distribution on a hyper-sphere  $\sum x_i^2=t$." 
 A: The point in $\#1$ is that the conditional density of $(X_1,\ldots,X_n)$ given the value of $X_1^2+\cdots+X_n^2$ does not depend on $\sigma.$
Suppose you're given that $X_1^2+\cdots+X_n^2 = r^2.$ The joint density of $(X_1,\ldots,X_n)$ is proportional (as a function of $(x_1,\ldots,x_n)$ to $$ e^{-(x_1^2+\cdots+x_n^2)/(2\sigma^2)}. $$ When restricted to the set on which that sum of squares is $r^2,$ that becomes $e^{-r^2/(2\sigma^2)}.$ Now the crucial point is that that value, $e^{-r^2/(2\sigma^2)}$, does not change as the point $(x_1,\ldots,x_n)$ moves about on the sphere $x_1^2+\cdots+x_n^2=r^2.$ So we have
$$
\int\limits_\text{sphere} e^{-r^2/(2\sigma^2)} \, dA = e^{-r^2/(2\sigma^2)} \times \text{surface area}.
$$
For this to be a density, we need to multiply it by a normalizing constant $c,$ so that
$$
\big(c e^{-r^2/(2\sigma^2)} \times \text{surface area} \big) = 1.
$$
Then the density is
$$
ce^{-r^2/(2\sigma^2)} = \left( \frac 1 {e^{-r^2/(2\sigma^2)} \times \text{surface area}} \right) e^{-r^2/(2\sigma^2)} = \cdots
$$
and this does not depend on $\sigma^2.$
I.e. the conditional distribution of $(X_1,\ldots,X_n)$ given $T$ does not depend on $\sigma^2.$
