Geometric Interpretation of Independence of Random Variables on the Standard Probability SpaceLet ( [ 0 , 1 ] , R ∩ [ 0 , 1 ] , λ ) be the standard probability space on [0,1] with λ, the Lebesgue measure. What do independent random variables "look like" in this space? Is there an easy geometric approach to creating independent functions on this space?

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Geometric Interpretation of Independence of Random Variables on the Standard Probability SpaceLet   (  [  0  ,  1  ]  ,      R    ∩  [  0  ,  1  ]  ,  λ  ) be the standard probability space on [0,1] with   λ, the Lebesgue measure. What do independent random variables &quot;look like&quot; in this space? Is there an easy geometric approach to creating independent functions on this space?

Anna Juarez · Accepted Answer

Step 1A collection of independent random variables (possibly infinite) defined on   (      I    ,            B                      I              ,  λ  ), where       I    :=  [  0  ,  1  ], can be constructed using a measure-preserving space-filling curve   φ  :      I    →      I    ×      I    ×  ⋯.Step 2The coordinate functions,   {      φ    i    }, are independent uniform random variables.

Let ([0,1],R cap [0,1], lambda) be the standard probability space on [0,1] with λ, the Lebesgue measure. What do independent random variables "look like" in this space? Is there an easy geometric approach to creating independent functions on this space?

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