Suppose I have that $Y_1, \ldots, Y_n$ are from a $Pois(\lambda)$ distribution, I am told that $\bar{Y} \pm 1.96\sqrt{\frac{\bar{Y}}{n}}$ a $95$% CI for $\lambda$.
However, I am not clear why. I know that for $n$ large, $\bar{Y} \sim N\left(\lambda, \frac{\lambda}{n}\right)$ approximately. However, how does this fact turn into the above? Can someone tell me where my derivations are incorrect below?:
$$ 0.95 = P\left(-1.96 \leq \frac{\bar{Y}-\lambda}{\sqrt{\frac{\lambda}{n}}} \leq 1.96\right) = P\left(\bar{Y}-1.96\sqrt{\frac{\lambda}{n}} \leq \lambda \leq \bar{Y}+1.96\sqrt{\frac{\lambda}{n}}\right) $$
However, at this point I CANNOT move the $\lambda$'s into the middle as it is not linear. Ideally, I would like $\lambda$ right in the middle, what am I doing wrong?