Define ????: ℝ → {0,1} as the characteristic function of ℤ+, the set of positive integers. That is, ????(x) = ????A (x), for the
set A of positive integers = { n | n ∈ ℤ+ }.
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Answer:
Step-by-step explanation:
The geometric distribution represents "the number of failures before you get a success in a series of Bernoulli trials. This discrete probability distribution is represented by the probability density function:"
Let X the random variable that measures the number os trials until the first success, we know that X follows this distribution:
In order to find the expected value E(1/X) we need to find this sum:
Lets consider the following series:
And let's assume that this series is a power series with b a number between (0,1). If we apply integration of this series we have this:
(a)
On the last step we assume that and
, then the integral on the left part of equation (a) would be 1. And we have:
And for the next step we have:
And with this we have the requiered proof.
And since we have that: