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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/.
- Suppose that N is uniformly distributed over the integers
1, 2, . . . , 10. Verify that
- 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.
- 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 + ½ y-½
for || considerably smaller than y >
0, write E[1/] as a series of terms
symmetrically expanded about the mean
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