Meaning of Independent Identically distributed random variables

AI Thread Summary
Independent Identically Distributed (IID) random variables are those that are both independent and share the same probability distribution. The term "identically distributed" implies that they have the same probability density function (pdf) and cumulative distribution function (cdf), which also means they possess the same mean and variance. In the example provided, the random variables N(0,1), N(2,4), and N(3,5) are independent but not identically distributed due to differing parameters, thus they are not IID. The key takeaway is that while independence is necessary, identical distribution is crucial for the IID classification. Therefore, for random variables to be considered IID, they must have identical statistical properties.
dionysian
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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?
 
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If they have the same pdf, then they by definition 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?
They are all normal, but since the parameters are different, the distribution functions are different. (Not IID, only I).
 
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Thanks.
 

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