normal approximation for discrete random variables

If it can be done, what are the minimum conditions for a normal approximation to be valid for discrete random variables. I know it can be done for a binomial if np and nq are larger than 10, but what about a case with not just simply success or failures?

For example,

x p(x)

0 0.2

1 0.3

2 0.5

mean is 1.3 and variance 0.61

if x is chosen 100 times could the total of x, lets call it Y, use normal approximation where Y ~N(100*1.3,sqrt(100*0.61)) ???

Cheers