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From an exponentially distributed population with parameter $\theta$ one draws a sample $X_1,...,X_n$ of size $n$. There is a wish to test the hypotheses $H_0: \theta=1$ against $H_1= \theta \neq 1$

A critical region $G$ is based on the likelihood ratio is a disjoint union $G=G_1 \cup G_2$ where ($c_1 < c_2$), $G_1=\{(x_1,...,x_n) \in \mathbb{R}_+^{n} : \overline{x} \leq c_1\}$ and $G_2=\{(x_1,...,x_n) \in \mathbb{R}_+^{n} : \overline{x} \geq c_2\}$. $\overline{x}=\frac{\sum_{k=1}^{n}x_k}{n}$

Problem Evaluate the level of significance of the test if one chooses $c_1=\frac{1}{2}$ and $c_2=1\frac{1}{2}$.

My attempt Let $\alpha= \mathbb{P}_{X_1,...,X_n}^{H_0}(G)= \mathbb{P}_{X_1,...,X_n}^{H_0}(G_1)+\mathbb{P}_{X_1,...,X_n}^{H_0}(G_2)=\mathbb{P}(\overline{x} \leq 0.5)+\mathbb{P}(\overline{x} \geq 1.5)$

but i don't understand how calculate the probability with exponential distribution if I have $n$ variables, could you help me with this?

StubbornAtom
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Aaron
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  • Note that the sum of independent exponential random variables has a Gamma (or Erlang) distribution, which might be useful to calculate those probabilities exactly. – LostStatistician18 Jun 07 '20 at 20:44
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    You can find the distribution of $\overline X$ (see https://math.stackexchange.com/q/655302/321264) directly or you may note that $2X_i\sim \chi^2_2$ independently under $H_0$ so that $2\sum_{i=1}^n X_i=2n\overline X\sim \chi^2_{2n}$ under $H_0$. – StubbornAtom Jun 07 '20 at 20:51

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