# Normal Approximation for the Poisson Distribution

Instructions: Compute Poisson probabilities using Normal Approximation. Please type the population mean $$\lambda$$ and provide details about the event you want to compute the probability for (notice that the numbers that define the events need to be integer. Also, if the event contains the sign "<", make sure to replace it by the equivalent event using $$\le$$. For example, if you need $$\Pr(X < 6)$$, compute instead $$\Pr(X \le 5)$$):

Population mean ($$\lambda$$)
Two-Tailed:
≤ X ≤
Left-Tailed:
X ≤
Right-Tailed:
X ≥

## Normal Approximation for the Poisson Distribution Calculator

More about the Poisson distribution probability so you can better use the Poisson calculator above: The Poisson probability is a type of discrete probability distribution that can take random values on the range $$[0, +\infty)$$.

When the value of the mean $$\lambda$$ of a random variable $$X$$ with a Poisson distribution is greater than 5, then $$X$$ is approximately normally distributed, with mean $$\mu = \lambda$$ and standard deviation $$\sigma = \sqrt{\lambda}$$.

A continuity correction needs to be used, so then to better adjust the approximation, so we use:

$\Pr(a \le X \le b) \approx \Pr(a - \frac{1}{2} \le X_{Normal} \le b + \frac{1}{2} )$ $= \Pr \left(\frac{a - \frac{1}{2} - \lambda}{\sqrt{\lambda}} \le Z \le \frac{b + \frac{1}{2} - \lambda}{\sqrt{\lambda}} \right)$

You can also use our calculator to compute the exact Poisson probabilities.

### Other normal approximations

A similar normal approximation is the normal approximation to the binomial distribution, which is actually more widely used than the one for the Poisson distribution.

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