# Minimum of i.i.d ~gamma random variables

• Pere Callahan
In summary: Your Name]In summary, the minimum of n i.i.d Gamma distributed random variables is not necessarily Gamma distributed, but can follow a generalized extreme value distribution. However, with a small enough scale parameter, it can be approximated by a Gamma distribution using the minimum order statistic.
Pere Callahan
Hi,

Another question...

I know that the minimum of n i.i.d $$\lambda$$-exponentially distributed random variables is again exponentially distributed (with parameter $$n\lambda$$). Is something similar true for $$\Gamma(k,\theta)$$ ...? that is, is the minimum of n i.i.d Gamma distributed random variables again Gamma distributed... or is it some other well known distribution?

I also know about the extreme-value theorem which might be of use if I were only interested in large n (which is actually the case) but an explicit distribution seems always better to me.

Thanks for any answers

-Pere

Hello Pere,

Thank you for your question. The minimum of n i.i.d Gamma distributed random variables is not necessarily Gamma distributed. It can follow a different distribution depending on the values of the shape and scale parameters (k and \theta). In general, the minimum of n i.i.d Gamma distributed random variables follows a distribution known as the generalized extreme value distribution, which is a type of extreme-value distribution.

However, if the scale parameter \theta is small enough, the minimum of n i.i.d Gamma distributed random variables can be approximated by a Gamma distribution with a different shape parameter k' and a smaller scale parameter \theta'. This approximation is known as the minimum order statistic and is commonly used in statistical analysis.

I hope this helps answer your question. Let me know if you have any further questions.

## 1. What does "i.i.d" stand for in the context of "Minimum of i.i.d ~gamma random variables"?

"i.i.d" stands for "independent and identically distributed". This means that each random variable in the set is independent from one another and follows the same probability distribution.

## 2. What is the significance of the minimum of i.i.d ~gamma random variables?

The minimum of i.i.d ~gamma random variables is often used in statistical analysis to represent the smallest value in a set of independent and identically distributed random variables that follow a gamma distribution. It can be used to model a variety of real-world phenomena, such as the minimum lifespan of a product or the time between events in a Poisson process.

## 3. How is the minimum of i.i.d ~gamma random variables calculated?

The minimum of i.i.d ~gamma random variables can be calculated by taking the minimum value from a set of random variables that follow a gamma distribution with the same shape and scale parameters. Alternatively, it can also be calculated using the cumulative distribution function (CDF) of the gamma distribution.

## 4. What is the relationship between the minimum of i.i.d ~gamma random variables and the exponential distribution?

The minimum of i.i.d ~gamma random variables can be seen as a special case of the exponential distribution, where the shape parameter is equal to 1. This means that the minimum of i.i.d ~gamma random variables can be used to model events that occur at a constant rate, such as radioactive decay or the waiting time between two consecutive events.

## 5. What are some real-world applications of the minimum of i.i.d ~gamma random variables?

The minimum of i.i.d ~gamma random variables can be applied in many fields, such as reliability engineering, queuing theory, and survival analysis. It can be used to model the time until failure of a product, the time between customer arrivals at a service center, or the minimum time until an event of interest occurs. It is also commonly used in actuarial science to model the minimum lifetime of an individual or the time until a certain event, such as retirement, occurs.

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