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Let

$\Omega = \{1,...,6\}^3$,

$X_k: \Omega \to \{1,...,6\}$ a Laplace-distributed random variable with $X_k(\omega_1,\omega_2,\omega_3)=\omega_k, k \in \{1,2,3\}$.

Define $S_1 := X_1+X_2, S_2:=X_2+X_3$.

I am having problems calculating the expected value of $\frac{S_1}{S_1+S_2}$, i.e. $$\Bbb{E}\left[\frac{S_1}{S_1+S_2}\right]$$

Any help is greatly appreciated!

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1 Answer 1

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$$\Bbb{E}\left[\frac{S_1}{S_1 + S_2} \right] = \Bbb{E}\left[ \frac{X_1 + X_2}{X_1 + 2X_2 + X_3}\right] = \sum_{\omega\in\Omega}\frac{\omega_1 + \omega_2}{\omega_1 + 2 \omega_2 + \omega_3} \underbrace{\Bbb{P}(X_1= \omega_1, X_2 = \omega_2, X_3 = \omega_3)}_{= \frac{1}{6^3}}\\ = \frac{1}{6^3}\sum_{i=1}^6\sum_{j=1}^6\sum_{k=1}^6 \frac{i + j}{i + 2 j + k} = \frac{1}{2}$$

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