When you have two of the same distribution, are they necessarily independent and identically distributed (i.i.d.)?

For example, when you have two from the normal distribution with the same expectation and variance, are they i.i.d.?


1 Answer 1


Two variables with the same distribution are necessarily identically distributed (that's what the "..i.d." part of "i.i.d." means -- that they have the same distribution).

However they are not necessarily independent. A pair of standard normal variates with correlation $\rho=0.8$ for example, are dependent, but identically distributed

scatterplot of a large sample from dependent but identically distributed variables

(in the diagram I show a pair that are bivariate normal, but that's simply for illustration; there's nothing that requires a pair of dependent identically distributed normals to be bivariate normal)

Here's two standard (and consequently identically distributed) exponential variates that are perfectly dependent, as we see in the third plot (in that the conditional distribution of each one on the other has no variance):

Histograms of two exponential random variables and a scatterplot showing they're monotonically (negatively) related


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