The Reproductive Property of Distributions


< Probability and Statistics Definitions

In statistics, the reproductive property is the result that if two or more independent random variables having a certain distribution are added, the resulting random variable has distribution of the same type as that of the summands. This is an important property to be aware of when working with distributions. Let’s take a closer look at how this property works.

Reproductive Property Examples

This graph shows how random variable is a function from all possible outcomes to numerical quantities and also how it is used for defining probability functions.
Niyumard, CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0, via Wikimedia Commons

Consider two random variables, X and Y, that have a certain distribution (e.g. normal, uniform, exponential, etc.). If we add these two random variables together, the resulting random variable Z will also have the same distribution as X and Y. This is true regardless of whether X and Y have the same distribution or not.

The reproductive property can be extended to more than two random variables as well. If we have three independent random variables, X, Y, and Z, each with its own probability distribution, and we add them all together, the resulting random variable W will also have the same distribution as X, Y, and Z.

This property is helpful because it allows us to make predictions about a sum of random variables even if we don’t know anything about their individual distributions. As long as we know the types of distributions involved, we can use the reproductive property to our advantage.

Conclusion

The reproductive property is a result in statistics that states if two or more independent random variables having a certain distribution are added, the resulting random variable has distribution of the same type as that of the summands. This is an important concept to be familiar with when working with distributions. Knowing about the reproductive property can help you make predictions about sums of random variables even if you don’t know anything about their individual distributions.


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