Meaning of Independent Identically distributed random variables

[dionysian]

I am a little fuzzy on the meaning of Independent Identically distributed random variables. I understand the independent part but still not 100% on the identically distributed part. I understand that identically distributed means they have the same pdf and cdf but does this mean that they have the same mean and variance?

For example if i have a sequence of random variables: N(0,1),N(2,4),N(3,5) and they are all independent are they IID?

 

[jav]

If they have the same pdf, then they by definition have the same mean and variance.

 

[mathman]

For example if i have a sequence of random variables: N(0,1),N(2,4),N(3,5) and they are all independent are they IID?

They are all normal, but since the parameters are different, the distribution functions are different. (Not IID, only I).

 

[dionysian]

Thanks.

 

[blue_raver22]

Independent identically distributed (IID) random variables are a fundamental concept in statistics and probability theory. Simply put, IID random variables are a collection of random variables that are independent of each other and have the same probability distribution. This means that each individual random variable in the collection follows the same probability distribution and is not affected by the values of the other variables.

In your example, the three random variables N(0,1), N(2,4), and N(3,5) are independent because the values of one variable do not affect the values of the others. However, they are not IID because they do not have the same probability distribution. The first variable has a mean of 0 and a variance of 1, the second has a mean of 2 and a variance of 4, and the third has a mean of 3 and a variance of 5.

IID random variables are useful because they allow us to make generalizations and predictions based on a sample of data. For example, if we have a sample of IID random variables, we can use the sample mean to estimate the population mean, and the sample variance to estimate the population variance. This concept is essential in many statistical analyses and is the basis for many important statistical tests and models.

In summary, IID random variables are a collection of independent random variables that have the same probability distribution, but they do not necessarily have the same mean and variance. Understanding this concept is crucial for accurately interpreting and analyzing data in many scientific fields.