Why a normal distribution would not give a good approximation to a specific discrete distribution?

Question

I have a discrete distribution $X$ with mean $56.87500$ and standard deviation $70.725$. I also have that the support of $X$ is contained in $[0, 8750]$, the maximum value of $X$ is $8750$ and $p(X=0)=(0.99)^{75}$. Why a normal distribution would not give a good approximation to X?

Thanks in advance for any help.

Answer 1

One major issue with a normal approximation is that a $N(57, 70)$ distribution gives a significant weigh---about 20%---to $(-\infty,0)$, which is outside your support. If your data is necessarily non-negative (maybe it counts something), then approximating with a distribution that gives 20% weight to negative numbers doesn't seem like a good idea.