Let $X_1, \dots, X_n$ be a random sample from a Poisson population with parameter $\lambda$ and define $Y = \sum_i X_i$. Y is sufficient for $\lambda$ and $Y \text{~} \text{Poisson}(n \lambda)$.
Identity: If $X \text{~ gamma($\alpha, \beta$)}$ for an integer $\alpha$, then for any $x$, $P(X \le x) = P(Y \ge \alpha)$ where $Y \text{~ Poission}(x/\beta)$.
By the identity above, $P(Y \le y_0 | \lambda) = P(\chi^2_{2(y_0+1)} \gt 2n\lambda)$.
Can someone explain how the identity above shows this? My thoughts were to put it into this form:
$P(X \le x) = P(Y \ge \alpha) \Rightarrow 1-P(X \le x) = 1-P(Y \ge \alpha)$
$\Rightarrow P(X \gt x) = P(Y \lt \alpha) \Rightarrow P(X \ge x) = P(Y \lt \alpha)$
But $P(Y \lt \alpha) \ne P(Y \le \alpha)$.