Suppose X is a real random variable with some density function f X ( x ). Let Y = g ( X ), where g ( . ) : R → R n .Is there a general way to represent the density f Y ( y )?For example, consider a simple case, X ∼ U n i f o r m ( 0 , 1 ). ( Y 1 , Y 2 ) = ( X + 1 , X + 2 ). Is there a closed form expression for the density of ( Y 1 , Y 2 ) in R 2 ?Edit:Dirac measures are permitted. In the above simple case, I presume the density could be f ( Y 1 , Y 2 ) ( y 1 , y 2 ) = { δ ( y 1 − y 2 + 1 ) 1 ≤ y 1 ≤ 2 and 2 ≤ y 2 ≤ 3 0 else

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Suppose   X is a real random variable with some density function       f    X    (  x  ). Let       Y    =      g    (  X  ), where       g    (  .  )  :      R    →            R        n  .Is there a general way to represent the density       f                  Y              (      y    )?For example, consider a simple case,   X  ∼  U  n  i  f  o  r  m  (  0  ,  1  ).   (      Y    1    ,      Y    2    )  =  (  X  +  1  ,  X  +  2  ). Is there a closed form expression for the density of   (      Y    1    ,      Y    2    ) in             R        2  ?Edit:Dirac measures are permitted. In the above simple case, I presume the density could be      f          (              Y        1            ,              Y        2            )        (      y    1    ,      y    2    )  =      {                            δ          (                      y            1                    −                      y            2                    +          1          )                                  1          ≤                      y            1                    ≤          2           and           2          ≤                      y            2                    ≤          3                                      0                                  else

grcalia1 · Accepted Answer

Your example was good, because it illustrates exactly why this can&#039;t be done in general. You cannot find a density function for that example, because the support of   (      Y    1    ,      Y    2    ), and hence the support of its density function   f (should it exist), would be the line segment   {  x  +  1  ,  x  +  2  :  x  ∈  [  0  ,  1  ]  }. Because that set has measure 0 in             R        2  , we would necessarily obtain       ∬    R    f        d    A  =  0 for any region   R.The issue is dimensionality; presuming g is well-behaved at all (e.g. continuous is enough to cause the problem), the support of   Y will necessarily be one-dimensional, and it will not be possible to furnish a density function in the higher-dimensional space.

Suppose X is a real random variable with some density function f X </msub> ( x

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