Probability: Proof of E[X/X+Y]=(X+Y)/2, X and Y are i.i.d. r.v.

In summary, the question raises some concerns about the validity of the given equation, but one approach to proving it would be to use a power series expansion of the reciprocal of a random variable.
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Homework Statement


Given that X and Y are i.i.d. r.v, and E[X]<infinity.
Any idea how to prove that E[X/X+Y]=E[Y/X+Y]=(X+Y)/2?


Homework Equations





The Attempt at a Solution


I know that E[X] or E[Y] are constants, how come it now says E[X/X+Y] = (X+Y)/2 which is a r.v.?
and for X and Y being independent, E[XY]=E[X]E[Y], but now it is E[X/X+Y], which I think it means E[X/(X+Y)], the upper and lower aren't independent, we cannot write E[X]/E[X+Y]...no idea where to begin with.

Any help is appreciated.
 
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  • #2
This seems weird to me, both for the reason you mention, that ##E[f(X,Y)]## can't be a function of the random variables, and also because you can easily show that

$$ E[X/(X+Y)] + E[Y/(X+Y)] = 1$$

for a normalized distribution. So the only way we can have ##E[X/(X+Y)] = E[Y/(X+Y)]## is if they are both 1/2.

Aside from those concerns, one way to approach the reciprocal of a random variable is through a power series. Namely, given a random variable ##X##, we can define a new random variable ##X'## by

$$ X = \bar{X} + X',~~~~ E[X']=0,$$

where ## \bar{X} = E[X]##. Then, if ##\bar{X}\neq 0##, we can use the geometric series to define

$$ \frac{1}{X} = \frac{1}{\bar{X}} \sum_{n\geq 0} (-1)^n \left( \frac{X'}{\bar{X}} \right)^n ,$$

which is convergent if ##|X'| \leq |\bar{X}|##. A similar construction would serve to define ##1/(X+Y)## or similar rational functions.

If you weren't already given a seemingly misleading answer to the question, this is the approach I would suggest.
 
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1. What is the formula for E[X/X+Y]?

The formula for E[X/X+Y] is (X+Y)/2, where X and Y are independent and identically distributed (i.i.d.) random variables.

2. How is the probability of E[X/X+Y] calculated?

The probability of E[X/X+Y] is calculated by taking the sum of all possible outcomes of the random variable X divided by the sum of all possible outcomes of both X and Y.

3. What does i.i.d. mean in the context of this formula?

I.i.d. stands for independent and identically distributed. This means that the random variables X and Y are independent of each other and have the same probability distribution.

4. Why is the formula for E[X/X+Y] important in probability?

The formula for E[X/X+Y] is important in probability because it allows us to calculate the expected value of a random variable in a given situation. It is a fundamental concept in probability theory and is used in various applications, such as in risk assessment and decision-making.

5. Can this formula be generalized for more than two i.i.d. random variables?

Yes, this formula can be generalized for any number of i.i.d. random variables. In general, the expected value of the sum of n i.i.d. random variables is equal to the sum of the expected values of each individual random variable divided by n.

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