Poisson distribution

Definitions

• Individual events occur at random and independently in a given interval.
• You know the mean number of occurrences in the interval or the rate of occurrences, and it’s finite. The mean number of occurrences is normally represented by $\lambda$.
  If $X$ follows a Poisson distribution with a mean of $\lambda$ occurrences per interval or rate, we write this as:
• $X\sim Po(\lambda )$

where

• X to represent the number of occurrences in the given interval
  

Expectation

• $E(X)=\lambda$

Variance

• $Var(X)=\lambda$

Combine Poisson Distributions

Approximate Poisson Distribution

Normal Distribution

If $X\sim Po(\lambda )$ and $\lambda >15$, you can use $X\sim N(\lambda ,\lambda )$ to approximate it.
If you’re approximating the Poisson distribution with the normal distribution, then you need to apply a continuity correction to make sure your results are accurate.

• Finding ≤ probabilities
  • if you’re using the normal distribution to find $P(X\le a)$, you actually need to calculate $P(X<a+0.5)$
• Finding ≥ probabilities
  • The value b extends down to $(b-0.5)$ on a continuous scale so you need to use a range of $P(X>b-0.5)$ to make sure that you include it.
• Finding “between” probabilities
  • $P(a-0.5<X<b+0.5)$

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