I have a question about the notation used in this problem please dont solve the problem for me.Show that | | X Y | | 1 = | | X | | 1 | | Y | | 1 for independent r.vs X and Y. Show further that if X and Y are also integrable, then E ( X Y ) = E ( X ) E ( Y )I don't understand what the difference between | | X | | 1 and E ( | X | ). I assume we're talking about the L 1 ( P ) norm? But then | | X | | 1 = ∫ Ω | X ( ω ) | d P ( ω ) = E ( | X | )So if this is finite, then E ( X ) exists therefore it is integrable already (so why "also"?) Therefore the first "show that" would follow from the second

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I have a question about the notation used in this problem please dont solve the problem for me.Show that       |        |    X  Y      |              |        1    =      |        |    X      |              |        1        |        |    Y      |              |        1   for independent r.vs   X and   Y. Show further that if   X and   Y are also integrable, then   E  (  X  Y  )  =  E  (  X  )  E  (  Y  )I don&#039;t understand what the difference between       |        |    X      |              |        1   and   E  (      |    X      |    ). I assume we&#039;re talking about the       L    1    (      P    ) norm? But then      |        |    X      |              |        1    =      ∫    Ω        |    X  (  ω  )      |    d      P    (  ω  )  =  E  (      |    X      |    )So if this is finite, then   E  (  X  ) exists therefore it is integrable already (so why &quot;also&quot;?) Therefore the first &quot;show that&quot; would follow from the second

Elbrinidr · Accepted Answer

Under independence       |        |    X  Y      |              |        1    =      |        |    X      |              |        1        |        |    Y      |              |        1   holds even if   E      |    X      |   or   E      |    Y      |   is infinity (by Tonelli&#039;s Theorem). But   E  (  X  Y  )  =  (  E  X  )  (  E  Y  ) holds if the expectations are finite.

I have a question about the notation used in this problem please dont solve the problem for me. Sho

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