The poisson distribution is a probability distribution for the number of occurrences of a rare event. In general, the probability of a variable X, which is modelled by the poission distribution, being equal to the value r is given by:
Where is the mean[/expr]$
The poission distrubution can be used to model situations where the occurences are independent and random and there is a known mean for the occurences
If the random variable X is modelled by the Poisson distribution with a mean then it is notated as:
The poission distribution can be used an approxmiation to the binnomial distribution, B(n, p) when n is large and p is small.
To mean of the poission distribution approximation of a binomial distribution is
The variance for the binomial distribution, B(n, p) is given by npq where q is (1 - p). Generally p is small enough such that q = 1 and hence the variance is just np. Therefore the variance is the same as the mean, np.
When finding a cumulative probability, for example P(X ≤ 4) you can either find and add together P(X = 0), P(X = 1) ... (P = 4). Alternatively you can use the poisson probability tables. Using these you find the value of (P ≤ 4) straight away by finding the value corresponding to the column of the mean and row of your value r.
To find, for example, P(X ≥ 5) simply find 1 - P(X ≤ 4)
The sum of two or more poisson distributions is also a poisson distribution, with the mean being the sum of the all the other means:
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