I'm trying to find a power function for a $X_1, ...,X_n$ ~ Exp($\theta$), where $H_0: \theta\geq \theta_0$ vs $H_a: \theta<\theta_0$; where, for test statistic X ~ $\sum X_i$, we would reject the null if $X \geq c$.
So far I came up with X being distributed as Gamma($n,\theta$). Because Gamma($\alpha, \beta$) doesn't have a closed form, I came up with $\theta X$ ~$Gamma(n,1$) to represent a CDF; my goal here is to find a power function and show that it's decreasing in $\theta$. Not sure what to do next or how to make sense of this CDF in the formula: $Pr(X \geq c|\theta$).