So, let's say that I was given two independent random variables ξ and η and was told that ξ has continuous distribution (basically, P ( ξ = c ) = 0 ∀ c ∈ R )How can I prove that in such case their sum also will have continuous distribution? Problem that I have here is that I know that if two variables are independent then we can say P ξ + η = P ξ ∗ P η , where ∗ stands for measures convolution operation, but I am nor really sure how to compute such a convolution. Can you provide some explanation or intuition on how to calculate such integrals? In my case I need to know integral like this: ∫ R 2 1 { c } ( x + y ) d P ξ ( x ) d P η ( y )And I have no rigorous way of showing that it is actually zero.

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So, let&#039;s say that I was given two independent random variables   ξ and   η and was told that   ξ has continuous distribution (basically,   P  (  ξ  =  c  )  =  0      ∀    c  ∈      R  )How can I prove that in such case their sum also will have continuous distribution? Problem that I have here is that I know that if two variables are independent then we can say       P          ξ      +      η        =      P    ξ    ∗      P    η  , where   ∗ stands for measures convolution operation, but I am nor really sure how to compute such a convolution. Can you provide some explanation or intuition on how to calculate such integrals? In my case I need to know integral like this:      ∫                            R                2                        1              {      c      }        (  x  +  y  )    d      P    ξ    (  x  )    d      P    η    (  y  )And I have no rigorous way of showing that it is actually zero.

Keegan Barry · Accepted Answer

By Fubini/Tonelli Theorem we have      ∫                            R                2                        1              {      c      }        (  x  +  y  )    d      P    ξ    (  x  )    d      P    η    (  y  )  =      ∫                  R                  P          ξ        (  c  −  y  )  d      P          η        y  =  0since       P          ξ        (  c  −  y  )  =  0 for every   y.

So, let's say that I was given two independent random variables ξ and &#x

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