# Independent and identically distributed (i.i.d)

In statistics, we commonly deal with random samples. A random sample can be thought of as a set of objects that are chosen randomly. Or, more formally, it’s “a sequence of independent and identically distributed (IID) random variables”.

In other words, the terms random sample and IID are basically one and the same. In statistics, we usually say “random sample,” but in probability it’s more common to use the term IID. So what does IID mean? Let’s break it down.

## Independent and identically distributed definition

IID stands for Independent and Identically Distributed Random Variables. A sequence of random variables is said to be IID if each random variable has the same probability distribution as the others and all are mutually independent.

Let’s unpack that a little bit. First, let’s talk about independency. Two random variables are independent if the occurrence of one random variable does not affect the probability of another random variable occurring. In other words, they are not related.

The second part, identically distributed, just means that each random variable has the same probability distribution function. So, in order for a set of random variables to be IID, they must be both independent and identically distributed.

But why is this important? Why does it matter if a set of random variables is IID?

Well, for one thing, IID is a necessary condition for many statistical procedures. In order for certain estimation methods to be valid, the data must be IID. Additionally, many theoretical results in probability and statistics require IID data in order to hold true.

So, if you’re working with a set of data that you think may be independently and identically distributed, great! You’re on your way to being able to use a variety of different statistical methods on your data set. And if you’re not sure whether or not your data meets the criteria for independence and identical distribution… well, that’s what statisticians are here for! We can help you figure it out.

## Conclusion

In conclusion, remember that IID stands for Independent and Identically Distributed Random Variables. A sequence of random variables is said to be IID if each random variable has the same probability distribution as the others and all are mutually independent. So next time you’re working with data sets in statistics, keep this concept in mind – it just might come in handy.

### One response to “Independent and identically distributed (i.i.d)”

1. […] are independent in 1785 [2].  Helmert’s transformation allows us to obtain a set of k – 1 new independent and identically distributed (i.i.d) normally distributed samples with μ = 0 and the same variance of the original […]