# Continuity correction, poisson

#### Tarly

##### New Member
Hi

There is something that puzzles me. I tried calculating a Poisson probability. Exactly, and with the normal distribution, using both the continuity and without the continuity correction. However the answer became much more exact if I did it not using continuity correction, do you see why?

It is a poisson distribution with parameter $$\lambda=168$$.
I want to calculate $$P(X \ge 196)$$.

I get that the exact answer is:
0.01876504
The answer using CLT and continuity correction :0.01693269
And without the continuity correction I get: 0.01862127

R-code:
Code:
1-ppois(195,168) #exact
1-pnorm((195.5-168)/sqrt(168)) #with correction
1-pnorm((195-168)/sqrt(168))   #without correction
Do you see why in this case it is best to not use continuity correction?

#### Dason

Your "without correction" method isn't actually correct. If you were to go without a correction you would just look at the area greater than or equal to 196 (exactly what the probability statement says).

Code:
> 1-ppois(195,168) #exact
[1] 0.01876504
> 1-pnorm((195.5-168)/sqrt(168)) #with correction
[1] 0.01693269
> 1-pnorm((196-168)/sqrt(168)) #without correction
[1] 0.01537678
So the correction is helping.

#### Tarly

##### New Member
Your "without correction" method isn't actually correct. If you were to go without a correction you would just look at the area greater than or equal to 196 (exactly what the probability statement says).

Code:
> 1-ppois(195,168) #exact
[1] 0.01876504
> 1-pnorm((195.5-168)/sqrt(168)) #with correction
[1] 0.01693269
> 1-pnorm((196-168)/sqrt(168)) #without correction
[1] 0.01537678
So the correction is helping.
I got 195 using this:

$$P(X \ge 196)=1-P(X <195)=1-P(\frac{X-168}{\sqrt{168}}<\frac{195-168}{\sqrt{168}})$$
$$\approx P(Z<\frac{195-168}{\sqrt{168}})$$

Did I do a mistake somewhere?