5.5 More on the square-root law We wish to explore the reasonableness of the approximation E[1/] 1/. We know from Section 5.8 that, in fact, E[1/] > 1/.

  1. Suppose that N is uniformly distributed over the integers 1, 2, . . . , 10. Verify that

  2. Now suppose that N's distribution is clustered more around its mean:
    Verify in this case that E[1/] 0.466, a result much closer to the desired approximation.

  3. Now suppose that E[N] is large and N's distribution is fairly symmetric about its mean E[N], which for simplicity we assume to be an integer. Using the square-root approximation
    (y + ) y + ½ y
    for || considerably smaller than y > 0, write E[1/] as a series of terms symmetrically expanded about the mean