How to keep the information of the confidence interval in the predictive step?

Suppose I have a sample $X}_{1},{X}_{2},\dots {X}_{n$. Each $X}_{i$ comes from a poisson distribution ${X}_{i}\sim Po\left(\lambda \right)$. The unknown paremeter $\lambda$ is inferred using the next confidence interval: $\lambda}^{\pm}=\stackrel{\u2015}{X}\pm {t}^{\cdot}\frac{\stackrel{\u2015}{X}}{\sqrt{n}$ where $\lambda}^{+$ is the upper bound and $\lambda}^{-$ is the lower bound, t∗ is the confidence expressed over t-student distribution.

I have two questions. The second question is actually the question I ask, the first is only to check my reasoning is right.

First: Taking into account the data come from Poisson distribution and taking into account the Central Limit Theorem, is it right the inference a did?It means: is it right to use the t-student approach? Or is it another better way to do?

Second: (That's the main question):

As I don't know the $\lambda$ parameter, I inferred it using confident intervals. Now I want to carry out some predictions of future events and calculate the associated probabilities, it means I wanto to know:

$prob\{{X}_{n+1}=K\}$. To do this: What number in the interval $[{\lambda}^{-},{\lambda}^{+}]$ I should take? Obviously, maximize the likelihood is the best option, so I would take $\stackrel{\u2015}{X}$. So the question is:

Is there a way to take into account the uncertainty in the probabilities of the prediction? or: Is there a way to keep the knowledge of the confident interval in the probabilities of the new prediction? or: At the moment in that you use an only number(point estimation), how to keep the uncertainty information?

To be clearer take into account these two examples. The same process but with two different interval confidence:

1) $[{\lambda}^{-},{\lambda}^{+}]=[6,8]$

$\stackrel{\u2015}{X}=7\to {X}_{n+1}\sim Po(\lambda =\stackrel{\u2015}{X})\to Prob\{{X}_{n+1}=6\}=0.449$

2) $[{\lambda}^{-},{\lambda}^{+}]=[1,13]$

$\stackrel{\u2015}{X}=7\to {X}_{n+1}\sim Po(\lambda =\stackrel{\u2015}{X})\to Prob\{{X}_{n+1}=6\}=0.449$

The first contain less uncertainty than the second, so how could I maintain the information of confident interval?

I hope I was clear in the question.