Let $X_1,...,X_{36} \overset{i.i.d}{\sim} \mbox{Poisson}(1)$ . Find $P( X< 1.1)$ by
(a) using the central limit theorem of $\bar{X}$;
(b) using the exact distribution of $\sum\limits_{i=1}^{36} X_i $.
I only need help with the approximate distribution of $\bar{X}$. I believe is distributed Normal as $N\sim(\lambda, \sqrt{\lambda})$ were $\lambda = 1$
Since I have $36$ $X$ vars I get $N\sim(1*36, \sqrt{1*36})$ $=$ $N\sim(36, 6)$
Or is it $N\sim(1, 1/36)$?
Am I up to a good start or am I lost?
EDIT:
Using $N\sim(1, 1/36)$ for (a) I got $P(Z < 0.6) = \mbox{pnorm}(0.6)$ Using $R$ software I got $.0726$
for (b) I'm doing the following:
$\bar{X} = \sum\limits_{i=1}^{36} X_i \sim \mbox{Poisson}(36)$ Hence $P(\bar{X} < 1.1) = P\left(\sum\limits_{i=1}^{36} X_i < 1.1\right) $
In $R$ I'm trying $\mbox{ppois}(1.1, 36)$ But I get an outrageous low number. If I however try $\mbox{ppois}(1.1, 1)$ I get $0.736$ which very close to the answer for (a).
Is there something else that needs to be done here?