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    Hi everyone! I'd really appreciate if anybody could help me understand this proof, I follow the first section ok, but then when they've let Y = X-a I don't understand why the function becomes f(y) in the integral rather than f(y+a) since x=y+a?

    Also im a bit confused by the manipulation of infinity and the integrals after that regarding how the negative signs on the limits and integral are changing.
    If anyone could try help me a bit I'd really appreciate it!

    Thank you
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    (Original post by F000sey)
    Hi everyone! I'd really appreciate if anybody could help me understand this proof, I follow the first section ok, but then when they've let Y = X-a I don't understand why the function becomes f(y) in the integral rather than f(y+a) since x=y+a?

    Also im a bit confused by the manipulation of infinity and the integrals after that regarding how the negative signs on the limits and integral are changing.
    If anyone could try help me a bit I'd really appreciate it!

    Thank you
    Name:  Screen Shot 2016-02-04 at 16.46.40.png
Views: 43
Size:  76.5 KB
    Goodness, that's very badly presented.

    First of all, the statement itself is false: there are plenty of well known examples of symmetric distributions where the expectation simply does not exist. So, the statement needs to be amended to say "if the expectation exists, then..."

    The second point is that the PDF of Y is not the same as the PDF of X - it's the PDF of X translated to be symmetric about the origin. So if you take your f(y+a) and call it f'(y) then you have the PDF of Y and the proof goes through.
 
 
 
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