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I Expected Value Question

  1. Feb 17, 2017 #1
    Hello all,
    I'm wondering if someone can offer some insight here: We have a random variable X, and it's expectation is called y.
    Can it be shown that
    1/y = E[1/X]
    ??
    Thanks
     
  2. jcsd
  3. Feb 17, 2017 #2

    mathman

    User Avatar
    Science Advisor
    Gold Member

    Not usually. Example: coin flip with heads (X=0) and tails (X=1). The expectation is 1/2. The expectation of 1/X is infinite.
     
  4. Feb 17, 2017 #3
    Thanks for your quick reply, mathman.
    I should have specified X is never zero.
     
  5. Feb 17, 2017 #4

    Dale

    Staff: Mentor

    Definitely not. Consider the random variable that takes values 1 and -1 with equal probability. The expectation is y=0 so 1/y is undefined. But E[1/X] is 0.
     
  6. Feb 17, 2017 #5
    Thanks Dale, as mentioned, I should have specified X is never zero, in fact it is always positive, and hence y>0.
    Thanks for your reply though!
     
  7. Feb 17, 2017 #6

    Dale

    Staff: Mentor

    It doesn't really matter, the point it that the relationship doesn't hold in general. That was just the easiest counterexample I could come up with in my head. Take pretty much any pair of numbers and you will get similar results.
     
  8. Feb 17, 2017 #7
    Dale:
    Right, I see what you're saying. If X={1,2,3} the y = 2. But then E[1/X] = 11/18.
    Could perhaps we say in general if y = E[X] that maybe 1/y < E[1/X] or perhaps even 1/y =< E[1/X] ?
    Thanks
     
  9. Feb 17, 2017 #8
    Wait a second. This is Jensen's Inequality: f(E[X])=<E[f(X)]
    In my case, we have f(X) = 1/X and y = E[X]. So I can say
    1/y =<E[1/X].
     
  10. Feb 17, 2017 #9

    Dale

    Staff: Mentor

    That could be. I think that 1/x is a convex function.
     
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