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This section is adapted from: S. K. Ross, Introduction to Probability Models, Third Edition, Academic Press, 1985, Chapter 3, pages 83-103.
PG-13
Let N be a random variable assuming positive integer values 1, 2, 3.... Let X_{i} be a sequence of independent random variables which are also independent of N with common mean E[X_{i}] independent of i. (This is a weaker hypothesis than independent, identically distributed random variables which is the typical case in applications.) Then
This is a special case of a more general theorem known as Wald’s equation.
Let N be a random variable assuming positive integer values 1, 2, 3.... Let X_{i} be a sequence of independent random variables which are also independent of N with common mean E[X] and common variance Var[X] which doesn’t depend on i. (This is a weaker hypothesis than independent, identically distributed random variables which is the typical case in applications.) Then
We know how to evaluate the second term. Let’s work on the first term. Condition on N.
Now work on the innermost conditional expectation and use that we are conditioning on N = n. Use the identity that E[Y ^{2}] = Var[Y ] + (E[Y ])^{2}, so (also using linearity of conditional expectations)
Notice that in the last step, we used that the variance of a sum of independent random variables is the sum of the variances. Now perform the outer expectation with respect to N.
Hence
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