# Let $X_1,X_2,X_3$ be iid. U($0,1$) random variables. Then what will be the value of $E(\frac{X_1+X_2}{X_1+X_2+X_3}$)? [closed]

Let $$X_1,X_2,X_3$$ be iid. U($$0,1$$) random variables. Then what will be the value of $$E(\frac{X_1+X_2}{X_1+X_2+X_3}$$) ?

## closed as off-topic by Namaste, Saad, StubbornAtom, Lord_Farin, DidJan 11 at 20:18

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So, convince yourself that random variables $$Y_i$$, defined through $$Y_i\triangleq \displaystyle\frac{X_i}{X_1+X_2+X_3}$$ for $$i =1,2,3$$ are identically distributed, due to symmetry. Hence, $$\mathbb{E}[Y_1]=\mathbb{E}[Y_2]=\mathbb{E}[Y_3]$$. Now, $$Y_1+Y_2+Y_3=1$$, implying that $$\mathbb{E}[Y_1+Y_2+Y_3]=1\implies \mathbb{E}[Y_i]=1/3$$ for every $$i$$.

From here, your answer is $$2/3$$.

• Downvoter, please step ahead and let me know why you did downvote my post. – Aaron Jan 17 at 15:47