Approximating $P(X \ge n)$ with CLT 


My question is in (d).  Suppose that we divide $X$ into many independent Poisson random variables each with $\lambda = 20/n$.  Then, we can set up:
$$P \left( \, \tfrac{X_1 + X_2 \ldots + X_n - 20}{\sqrt {20}} \ge \tfrac{6}{\sqrt {20}} \, \right) \ \ = \ \ 0.0901$$
As it turns out, I need to adjust the boundary value to $5.5/\sqrt{20}$:
$$P \left( \, \tfrac{X_1 + X_2 \ldots + X_n - 20}{\sqrt {20}} \ge \tfrac{5.5}{\sqrt {20}} \, \right) \ \ = \ \ 0.1093$$
I can imagine that the purpose is to fit the entire bar for the value $6$ into the area of consideration.  But is it customary to use $0.5$ as adjustment?  For example, if I am approximating $P(X = 26)$, I don't see what is wrong with the following, even though $0.5$ may be most natural:
$$P(25.5 \le X \le 26.5) \ \ \ \ \ P(25.75 \le x \le 26.75) \ \ \ \ \ P(25.9 \le x \le 26.9) \ \ \ \ \ \text{or ...}$$
 A: First, note that
$$
\Pr(X \geq 26) + \Pr(X \leq 25) = 1
$$
However,
$$
\Pr\left(\frac{X_1 + \cdots + X_n - 20}{\sqrt{20}} \geq \frac{6}{\sqrt{20}}\right) + \Pr\left(\frac{X_1 + \cdots + X_n - 20}{\sqrt{20}} \leq \frac{5}{\sqrt{20}}\right) \neq 1
$$
following your notations. The problem arises because you are attempting to approximate a discrete distribution using a continuous distribution and as an easy way to address the problem, one can use the middle value as the cut point; that is, we use
$$
\Pr\left(\frac{X_1 + \cdots + X_n - 20}{\sqrt{20}} \geq \frac{5.5}{\sqrt{20}}\right)\quad \text{and}\quad \Pr\left(\frac{X_1 + \cdots + X_n - 20}{\sqrt{20}} \leq \frac{5.5}{\sqrt{20}}\right)
$$
instead to approximate $\Pr(X \geq 26)$ and $\Pr(X \leq 25)$. It should be noted however the middle point, though natural, may be not the best cut point since other cut points may lead to better approximation results (by some approximation measures). It needs some research to decide the best one but I think the middle one would be good usually.
