For a discrete rv X taking values in {0, 1, 2, 3, }, we shall derive the
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For a discrete rv X taking values in {0, 1, 2, 3, …}, we shall derive the following alternative formula for the mean:
a. Suppose for now the range of X is {0, 1, …, N} for some positive integer N. By re-grouping terms, show that
b. Re-write each row in the above expression in terms of the cdf of X, and use this to establish that
c. Let N → ∞ in part (b) to establish the desired result, and explain why the resulting formula works even if the maximum value of X is finite.
d. Let X have a geometric distribution with parameter p. Use the cdf of X and the alternative mean formula just derived to determine μX.
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Modern Mathematical Statistics With Applications
ISBN: 9783030551551
3rd Edition
Authors: Jay L. Devore, Kenneth N. Berk, Matthew A. Carlton
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