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I'm reading Stochastic Calculus for Finance II: Continuous-Time Models by Steven Shreve and I don't understand how he went from the equation on the left to the middle one. If it helps, this section is proving that the distribution of a scaled random walk converges to the normal distribution.

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    $\begingroup$ isn't that becuase X is either +1 or -1 with 50% probability, so the step you are highlighting is the discrete expectation of these two outcomes? $\endgroup$
    – Attack68 ♦
    Oct 17, 2020 at 7:15

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$X_j$ can be either 1 or -1 with 50% probability each. So this step is just applying the expectation to both possible cases.

See definition of the Expectation... \begin{align} {\mathbb E}\bigl[ X \bigr] = \sum_i i \cdot p(x = i) \end{align}

It's the sum over all possibilities of the probability of getting that value (both ${\frac 1 2}$ in your case) multiplied by the value

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