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

In summary, the conversation discusses the relationship between a random variable X and its expectation, y. The main question is whether or not it can be shown that 1/y is equal to the expectation of 1/X. However, it is shown that this relationship does not hold in general and is dependent on the values of X. It is also mentioned that this is an example of Jensen's Inequality, where the function 1/X is convex.
  • #1
Steve Zissou
49
0
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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  • #2
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.
 
  • #3
Thanks for your quick reply, mathman.
I should have specified X is never zero.
 
  • #4
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.
 
  • #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!
 
  • #6
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.
 
  • #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
 
  • #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].
 
  • #9
That could be. I think that 1/x is a convex function.
 
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What is Expected Value?

Expected value is a statistical concept that represents the average outcome of a random variable over a large number of trials. It is calculated by multiplying the probability of each possible outcome by its respective payoff and summing all of these values together.

Why is Expected Value Important?

Expected value is important because it allows us to make informed decisions in situations where there is uncertainty. By calculating the expected value, we can determine the most likely outcome and use this information to guide our actions.

How is Expected Value used in Decision Making?

Expected value is used in decision making by comparing the expected value of different options and choosing the one with the highest expected value. This helps us to make rational decisions that maximize our potential payoffs.

What are the Limitations of Expected Value?

Expected value does not take into account the potential risks and uncertainties associated with an outcome. It assumes that all possible outcomes are equally likely to occur, which may not always be the case in real-world situations.

Can Expected Value be Negative?

Yes, expected value can be negative. This means that the potential payoff for a particular decision is less than the cost or investment required. In this case, it may be more beneficial to choose a different option with a higher expected value.

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