# What does $\sum_{\{\textbf{x}:T(\textbf{x})=t\}} f(\textbf{x}; \theta)$ mean?

This is in the following context:

$$q(t;\theta) = P(T=t;\theta) = \sum_{\{\textbf{x}:T(\textbf{x})=t\}} f(\textbf{x}; \theta)$$

Where $$T=T(\textbf{X})$$ is a statistic, $$q(t; \theta)$$ is the pmf of $$T$$, $$X_1, X_2, ..., X_n$$ is a random sample and $$f(\textbf{x}; \theta)$$ is the pmf of $$\textbf{X}$$.

This is simply summing the probabilities of all combinations of $$X$$ that lead to $$T(X)=t$$, since it asks for $$P(T(X)=t)$$. For example, $$T=X_1+X_2$$: a statistic with respect to the outcome of the two dice, $$X_1,X_2$$ are face values, and $$t$$, e.g. $$10$$ is the sum of them. You'll have the following sum, $$\sum_{X_1+X_2=t}P(X_1,X_2)$$ that resembles your general sum.

This simply means sum up the values of the function $$f(\textbf{x}; \theta)$$ over all values of $$\textbf{x}$$ where the test statistic $$\textbf{T(x)}=t$$