What is the reproductive property

The reproductive property of a probability distribution is the property that the sum of several independent random variables following the same distribution follows the original distribution. Among the many probability distributions, some probability distributions have the reproductive property.

Probability distributions with reproductive property

The following probability distributions are known to have the reproductive property.

• Normal distribution
• Binomial distribution
• Poisson distribution
• Gamma distribution
• Chi-Square distribution

Normal distribution

Suppose that the random variables X and Y are independent of each other, each following a normal distribution as follows

X \sim N(\mu_1, \sigma^2_1),\quad Y \sim N(\mu_2, \sigma^2_2)

In this case, X + Y follows the following normal distribution.

X + Y \sim N(\mu_1 + \mu_2, \sigma^2_1 + \sigma^2_2)

Binomial distribution

Suppose that the random variables X and Y are independent of each other, each following a binomial distribution as follows

X \sim B(n_1, p), \quad Y \sim B(n_2, p)

In this case, X + Y follows the following binomial distribution.

X + Y \sim B(n_1 + n_2, p)

Poisson distribution

Suppose that the random variables X and Y are independent of each other, each following a Poisson distribution as follows

X \sim Po(\lambda_1), \quad Y \sim Po(\lambda_1)

In this case, X + Y follows the following Poisson distribution.

X + Y \sim Po(\lambda_1 + \lambda_2)

Gamma distribution

Suppose the random variables X and Y are independent of each other, each following a gamma distribution as follows

X \sim Ga(\alpha_1, \beta), \quad Y \sim Ga(\alpha_2, \beta)

In this case, X + Y follows the following gamma distribution.

X + Y \sim Ga(\alpha_1 + \alpha_2, \beta)

Chi-square distribution

Suppose that the random variables X and Y are independent of each other, each following a chi-square distribution as follows

X \sim \chi^2(n_1),\quad Y \sim \chi^2(n_2)

In this case, X + Y follows the following chi-square distribution.

X + Y \sim \chi^2(n_1 + n_2)