probability or statistics - TransformedDistribution using $k$ iid random variables

How do I create a TransformedDistribution that uses $k$ independent identically distributed (i.i.d.) random variables?

For example, I can derive a chi-squared distribution with 2 degrees of freedom like so:

 In[1]:= TransformedDistribution[x^2 + y^2,
           {x \[Distributed] NormalDistribution[], y \[Distributed] NormalDistribution[]}]
Out[1]:= ChiSquareDistribution[2]

Given an arbitrary integer $k$ , how do I similarly derive a chi-squared distribution with $k$ degrees of freedom using TransformedDistribution?

Answer

In this case, you can simply use ChiSquareDistribution[k], but in the general case, of the sum of k variables distributed as dist:

iid[k_, dist_] := TransformedDistribution[
                    Sum[a[i], {i, k}],
                    Table[Distributed[a[i], dist], {i, k}]
                  ]

Comments

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