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 μ_X.

Transcribed Image Text:

0=x [(x) - 1] = xrl