Expected Value of Random Variable X: Solving for E[1/X]

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Discussion Overview

The discussion revolves around the expected value of a random variable X and whether the relationship 1/y = E[1/X] holds, where y is the expectation of X. Participants explore this concept through various examples and counterexamples, focusing on the implications of X being positive and the application of Jensen's Inequality.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions if 1/y = E[1/X] can be shown, where y is the expectation of X.
  • Another participant argues that this relationship does not usually hold, providing an example where X can be zero.
  • A subsequent reply clarifies that X is never zero, indicating that it is always positive, thus y > 0.
  • Another example is presented where a random variable takes values 1 and -1, leading to an undefined expectation for y but a defined expectation for E[1/X].
  • One participant suggests that the relationship does not hold in general, indicating that similar counterexamples can be found.
  • A participant provides a specific case where X takes values {1,2,3}, calculating y and E[1/X] to illustrate the discrepancy.
  • Another participant introduces Jensen's Inequality, suggesting that 1/y ≤ E[1/X] may hold under certain conditions.
  • There is a consideration that the function 1/x is convex, which relates to the application of Jensen's Inequality.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether 1/y = E[1/X] holds, with multiple competing views and examples presented that illustrate the complexity of the relationship.

Contextual Notes

Participants note that the expectation of X being positive is a critical assumption, and the discussion includes various counterexamples that highlight the limitations of the proposed relationship.

Steve Zissou
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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
 
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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.
 
Thanks for your quick reply, mathman.
I should have specified X is never zero.
 
Steve Zissou said:
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
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.
 
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!
 
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.
 
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
 
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].
 
That could be. I think that 1/x is a convex function.
 
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