# Hypothesis testing: Calculating Type I and II error

Let $$X_i \sim Pois(\lambda)$$, $$i=1, ...,10$$. We have the two hypothesis:

$$H_0 : \lambda = 0.1$$ and $$H_{\alpha} : \lambda = 0.5$$. We reject $$H_0$$ if $$\sum_{i=1}^{10} X_i \geq 3$$.

I am trying to calculate the significance level $$\alpha$$ and the power of test $$1 - \beta$$. I was thinking about calculating the type 1 and 2 error, but I think my results aren't correct:

$$\alpha = \mathbb{P} [\sum_{i=1}^{10} X_i \geq 3 | H_0] = \sum_{k=3}^{10} \frac{0.1^k}{k!} \approx 0.000155$$

and

$$1 - \beta = 1 - \mathbb{P} [\sum_{i=1}^{10} X_i \leq 2 | H_{\alpha}] = 1 - 0.9856123$$, but this result seems not to be really plausible...

Let $$T=\sum_{i=1}^{10} X_i$$. Supposing that the $$X_i$$ are independent, under the null, it is the case that $$T\sim \text{Poisson}(1)$$ (where we are using the fact that in general if $$X\sim \text{Poi}(\lambda_1)$$ and $$Y\sim\text{Poi}(\lambda_2)$$ and $$X, Y$$ are independent then $$X+Y\sim \text{Poi}(\lambda_1+\lambda_2)$$.
In particular, the probability of type I error is given by $$\alpha=P(T\geq 3)=1-P(T=0)-P(T=1)-P(T=2)$$ where $$T\sim \text{Poi}(1)$$. Similarly, the power of the test, i.e. the probability that we reject the null given that it is false is given by $$1-\beta=P(T\geq 3)=1-P(T=0)-P(T=1)-P(T=2)$$ where $$T\sim \text{Poi}(5)$$