So, let's say that I was given two independent random variables ξ and η and...

pouzdrotf

pouzdrotf

Answered

2022-07-04

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.

Answer & Explanation

Keegan Barry

Keegan Barry

Expert

2022-07-05Added 18 answers

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 = 0
since P ξ ( c y ) = 0 for every y.

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