Use the central limit theorem to approximate a probability of a sum of Poisson random variables.

Question

$X_1, \ldots, X_{10}$ are independent $\text{Poisson}(1)$. I need to find $P(X_1 +\cdots + X_{10}\geq 15) $. I think I would like to use the central limit theorem to say that $$P(\frac{\bar{x}-10}{\sqrt{10}}\geq\frac{15-10}{\sqrt{10}})=P(Z\geq1.581)=1-P(Z<1.581)\approx.0571$$

Is this the correct approach or have I done something wrong?

Answer 1

The sum of independent Poisson random variables is Poisson. So that much is rock solid. However, the normal approximation to Poisson works best for large $\lambda$ and 10 is not quite large enough for most applications of the CLT.

Nowadays software simplifies exact computation. The exact result (to four places) from R statistical software is $0.0835 \ne 0.0571:$

1 - ppois(14, 10)
## 0.08345847

Your normal approximation could be improved by a "continuity correction" and interpolation in using printed normal tables (or using software for the normal value). You can read about the continuity correction, or look at the figure to see why you want the approximating area to lie under the normal curve and above 14.5. Notice that $P(X \ge 15) = P(X > 14.5) = P(X > 14)$ because the discreteness of the Poisson random variable.

1 - pnorm(14.5, 10, sqrt(10))
## 0.07736446