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Suppose $X_1,X_2,...$ is an infinite sequence of i.i.d random variables and let $N$ be a positive integer valued r.v independent from $\{ X_i \} $. Let $S_n = \sum_{i=1}^n X_i $ and $S_N = X_1 + ... + X_N $ with convention that $S_N = 0 $ if $N = 0 $. if $\mathbb{E} \{ N \} < \infty $ and $\mathbb{E} \{ | X_j| \} < \infty $, can we conclude that

$$ \mathbb{E} \{ S_N \} = \sum_{n=0}^{\infty} \mathbb{E} \{S_n \} P(N = n ) $$

??

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    $\begingroup$ I don't see any difference between $S_n$ and $S_N$. $\endgroup$
    – GFauxPas
    Apr 6 '15 at 18:49
  • $\begingroup$ @GFauxPas $n$ is a simply an integer, $N$ is a random variable. $\endgroup$ Apr 8 '15 at 13:21
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Hints:

  1. Note that $$S_N = \sum_{j \in \mathbb{N}} S_j 1_{\{N=j\}}.$$ Taking expectations on both sides yields $$\mathbb{E}(S_N) = \mathbb{E} \left( \sum_{j \in \mathbb{N}} S_j 1_{\{N=j\}} \right).$$

  2. Conclude from the monotone convergence theorem and the assumed independence that $$\mathbb{E} \left(\sum_{j \in \mathbb{N}} |S_j| 1_{\{N=j\}} \right) \leq \sum_{j \in \mathbb{N}} j \mathbb{E}(|X_1|) \mathbb{P})(N=j) = \mathbb{E}(|X_1|) \cdot \mathbb{E}(N)<\infty.$$

  3. By step 1,2 and the dominated convergence theorem, $$\mathbb{E}(S_N) =\sum_{j \in \mathbb{N}} \mathbb{E}(S_j 1_{\{N=j\}}) = \sum_{j \in \mathbb{N}} \mathbb{E}(S_j) \mathbb{P}(N=j).$$
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