The Reproductive Property of Distributions

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Reproductive Property of Distributions

The reproductive property of distributions tells you that when you add two or more independent random variables that belong to the same family of probability distributions, the result will have the same distribution as the original variables. For example, the sum of a set of normal random variables is itself a normal.

“Independence” means that the occurrence of one variable does not influence the probability or outcome of the other variable(s). In other words, the random variables are not related and their outcomes do not depend on each other. 

More formally, we can say that if we sum two or more independent random variables, their probability density functions (pdfs) are distributed according to that pdf– albeit with different parameters.

The reproductive property holds for distributions that are closed under addition i.e., where the sum of any two random variables that have the same distribution also has the same distribution). This includes many popular distributions such as:

• Binomial distribution,
• Chi-square distribution,
• Exponential distribution,
• Normal (Gaussian) distribution,
• Poisson distribution.

“Closed under addition” refers to the general algebraic property of a set, ensuring that the sum of its elements remains within the set.

Reproductive Property Examples

If two random variables, X and Y, have a certain probability distribution (e.g., exponential, normal, or Poisson), their additive distribution Z will have the same distribution as X and Y. However, their properties may be different.

For example, two random variables, X and Y, with a uniform distribution ranging from 0 to 1, are added to create a new random variable, Z. The distribution of Z is also uniform, but with a range of 0 to 2. This happens because the sum of two uniformly distributed random variables also follows a uniform distribution. In other words, the uniform distribution has the Reproductive Property.

Also:

• The mean (µZ) will be the sum of the means of X and Y (µX + µY). So if distribution X has a mean of 3 and distribution Y has a mean of 4, then Z will have a mean of 7.
• The variance of Z (σ²Z) will be the sum of the variances of X and Y (σ²X + σ²Y),

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, a resulting random variable W will also have the same distribution as X, Y, and Z.

Note on the reproductive property of exponential random variables

The sum of independent exponential random variables follows a gamma distribution, but we still can say that the exponential distribution has the reproductive property. The gamma distribution is a more general distribution than the Erlang distribution (a special case of the gamma distribution); However, if the rate parameters of the exponential distributions are equal, the sum will also be an Erlang distribution. Rate parameters describe exponential distributions or event rates in a Poisson process.

The addition of independent exponential random variables results in a gamma distribution, where the shape parameter is determined by the count of exponential random variables being combined, and the scale parameter is the sum of their rate parameters. If all the exponential random variables have identical rate parameters, the resulting distribution will be an Erlang distribution, with its shape parameter equivalent to the number of exponential random variables being added together.

References

1. Yost, G. P. Lectures on probability and statistics. Presented as a course on Statistics, Imperial College, London, O.K., January-April 1983