Question

If X & Y are symmetric random variables, then show that E(X/X+Y)= 1/2

Answer 1

Answer on Question #46919 – Math – Statistics and Probability

If X & Y are symmetric random variables, then show that $E\left(\frac{X}{X + Y}\right) = \frac{1}{2}$.

Solution

Rename X as Y and Y as X. Then, by symmetry,

$$E \left(\frac {X}{X + Y}\right) = E \left(\frac {Y}{Y + X}\right) = E \left(\frac {Y}{X + Y}\right) .$$

Now,

$$E \left(\frac {X + Y}{X + Y}\right) \equiv 1.$$

But the left side is

$$E \left(\frac {X}{X + Y}\right) + E \left(\frac {Y}{X + Y}\right) = 2 E \left(\frac {X}{X + Y}\right), \text{ by (1) above}.$$

Then $2E\left(\frac{X}{X + Y}\right) = 1$. Therefore,

$$E \left(\frac {X}{X + Y}\right) = \frac {1}{2},$$

as was to be shown.

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