how the conditional distribution of $(\sum y_i^2 |\sum y_i, )$ form an exponential family Suppose a sample from the normal distribution, the canonical statistic is: $\boldsymbol{t(y)} = (\sum y_i, \sum y_i^2) = (v,u)$.  The distribution  $f(u|v;\theta)$ forms an exponential family.  
In my book the explanation to this is:
Given $\sum y_i = n\bar{y}$, $u$ differs only by a constant from $\sum(y_i - \bar{y})^2 = (n-1)s^2$ (this constant is $(-n\bar{y})$). Thus, it is enough to characterize  the distribution of $(n-1)s^2$ and this distribution is proportional (by $\sigma$)  to $\chi^2(n-1)$. From the explicit form of  the $\chi^2$ it is easily seen that the conditional distribution forms an exponential family.


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*$u$ differs only by a constant from $\sum(y_i - \bar{y})^2 = (n-1)s^2$. Why is it:


enough to characterize  the distribution of $(n-1)s^2$

is the constant ($-n\bar{y}$) irrelevant and why?


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*$(n-1)s^2$ is proportional to $\sigma^2 \chi(n-1) $



yes ok, but what does that say about  $u$? and how the conditional
  distribution forms an exponential family?

 A: I agree that the explanation as given is somewhat unclear, assuming some extra knowledge on the part of the reader.
First, constants are usually irrelevant in the math, because, writing loosely and at a high level, they don't change the shape of the probability distribution, they just position (or shift) it on the real line.  Think of a Normal variate - we don't have a different theory for Normal distributions with mean 0.282983 than we do for Normal distributions with mean 3279 or mean 0.  Because the shapes are the same, the properties are the same, at least for properties that don't depend on the location of the distribution (hence the "usually".)  With respect to membership in the exponential family, adding or subtracting a known constant from the random variable doesn't change whether or not the associated distribution is a member of the exponential family.
Second, since $u = (n-1)s^2 +n\bar{y}^2$, and $n\bar{y}^2$ is a constant (as we are conditioning upon $\bar{y}$), knowing that $(n-1)s^2 \propto \sigma^2\chi^2(n-1)$, again writing loosely, means we know that the distribution of $u$  is just a shifted and rescaled $\chi^2$ distribution.  And, since the $\chi^2$ distribution is a member of the exponential family, it follows that the conditional distribution of $u$ is a member of the exponential family.
