Bounded Probability Distribution

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What is a bounded probability distribution?

A bounded probability distribution is one that is limited to lie between two specified values. Outside of these values, the probability of an event is zero. Some examples of bounded distributions include:

Other notable bounded distributions include the arcsine, Bates, Kumaraswamy, raised cosine, Topp-Leone and von Mises distributions.

Bounded distributions have “natural” constraints. For example, the categorical distribution ranges from zero categories to n categories; it would make no sense to have minus 1 or an infinite number of categories in this distribution. On the other hand, unbounded distributions do not have constraints and theoretically can extend from negative infinity to positive infinity — although many probability distributions peter out before they reach that theoretical range.

Partially unbounded distributions are constrained at one end. For example, the Poisson distribution is partially bounded as it is constrained to be non-negative.

Why use a bounded probability distribution?

When analyzing real life phenomena, these usually involve a finite range of changes. These finite changes usually give rise to bounded domain distributions [1]. For example, an upper bound can be very helpful in analyzing the annual stream flow and annual rainfall data [2].

As another example, if we want to model the heights of NBA basketball players, it makes no sense to include small values < 5′ 3″ (the height of Muggsy Bogues, the shortest basketball player in history) because the the average height of NBA players is about 6’6”. Thus, we can include a partial bound in the model from 5′ 3″ < ∞.

Sometimes, it may be necessary to restrict unbounded and partially bounded distributions to avoid generating nonsensical values. For example, if you’re modeling the number of sales, you’ll want to exclude negative numbers from the distribution.

Truncated vs bounded probability distribution

While a bounded probability distribution is confined between two specified values, a truncated probability distribution is usually created by restricting the domain of an unbounded distribution. For instance, we can truncate the normal distribution to only allow values between 0 and 1. This leads to a new distribution called the truncated normal distribution.

bounded probability distribution example graph of a truncated normal
A lower truncated standard normal distribution.


To put this another way, a bounded probability distribution has a natural or built in finite range. For example, the range of a binomial distribution is between 0 and n trials. But when we force bounds on an unbounded distribution, such as the normal distribution, it is called a truncated distribution. Although the truncated distribution is cut-off, it can still take on an infinite number of values, albeit with a probability of zero.


  1. Bakouch, H.S., Hussain, T., Chesneau, C. et al. A notable bounded probability distribution for environmental and lifetime data. Earth Sci Inform 15, 1607–1620 (2022).
  2. Phien HN, Ajirajah TJ (1984) Applications of the log Pearson type-3 distribution in hydrology. J Hydrol 73(3-4):359–372.

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