Meaning of Independent Identically distributed random variables

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SUMMARY

Independent Identically Distributed (IID) random variables must share the same probability density function (pdf) and cumulative distribution function (cdf), which implies they also have identical means and variances. In the provided example, 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 distinction between independence and identical distribution is crucial in statistical analysis.

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  • Understanding of probability density functions (pdf)
  • Knowledge of cumulative distribution functions (cdf)
  • Familiarity with normal distributions
  • Basic concepts of independence in probability theory
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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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