1
$\begingroup$

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.

enter image description here

$\endgroup$
1
  • 5
    $\begingroup$ 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

1 Answer 1

4
$\begingroup$

$X_j$ 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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.