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Truncated Probability Distribution


Truncated distributions are those distributions for which some of the domain values have been cut off as they cannot be realistically attained in some given real-life application.

We need to make changes in the original probability distribution function to account for the truncation as the total probability must still add up to one irrespective of whether some values have been truncated or not.

For example, to model waiting times at a bus stop, we may use the exponential distribution truncated at some finite value because it might be unrealistic to expect an indefinitely large waiting time.

Types of Truncated Probability Distributions:

Some examples of truncated distributions are the truncated normal distribution, the truncated binomial distribution, and the truncated Poisson distribution.

Truncated Binomial Distribution

The zero truncated binomial distribution is the binomial distribution whose probability distribution function has the value at zero removed. It is used in the analysis of counting type data where the value zero cannot possibly be attained.

Probability Distribution Function of truncated binomial distribution
Mean and Variance of Truncated Binomial Distribution

Truncated Poisson Distribution

The zero truncated Poisson distribution is the Poisson distribution whose probability distribution function has the value at zero removed. An example application would be the number of dishes a customer at a restaurant might order to eat. In this case, it is reasonable to assume that the customer would not go to the restaurant if he had no dishes to order.

Probability Distribution Function of truncated Poisson distribution
Mean and Variance of Truncated Poisson Distribution

Truncated Normal Distribution

The truncated normal distribution is obtained by the usual normal distribution by removing extremely high or low values. If values are removed from both sides it is a two-sided truncation otherwise it is a one-sided truncation.

Such truncations are usually done in cases where it is impractical to assume that the given value can be extremely high or extremely low because the usual normal distribution extends to infinity on both sides.

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