# Comparison of Distribution Functions

DistributionFunctional FormMeanStandard Deviation

### Poisson

If the probability of an event is p = and there are n = events, then the probability of that event being observed is: For these conditions, the mean number of events is and the standard deviation is .
 Distribution Value of x Probability
Binomial x 10^

### Note!

This calculation must evaluate the factorials of very large numbers if the number of events is large. A sum of logarithms was used to evaluate the factorials, and use was made of the fact that there is a quotient of factorials which can be reduced. Still, the factorials can get too large. Use of the Gaussian distribution may be necessary for large values of n.
Index
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# Binomial Distribution Function

 Distribution Functional Form Mean Standard Deviation Binomial

The binomial distribution function specifies the number of times (x) that an event occurs in n independent trials where p is the probability of the event occurring in a single trial. It is an exact probability distribution for any number of discrete trials. If n is very large, it may be treated as a continuous function. This yields the Gaussian distribution. If the probability p is so small that the function has significant value only for very small x, then the function can be approximated by the Poisson distribution.

If the probability of an event is p = and there are n = events, then the probability of that event being observed times is x 10^. For these conditions, the mean number of events is and the standard deviation is .

### Note!

This calculation must evaluate the factorials of very large numbers if the number of events is large. A sum of logarithms was used to evaluate the factorials, and use was made of the fact that there is a quotient of factorials which can be reduced. Still, the factorials can get too large. Use of the Gaussian distribution may be necessary for large values of n.
Index

Distribution functions
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