MINIMUM VARIANCE UNBIASED ESTIMATION OF THE SCALE PARAMETER OF EXPONENTIAL DISTRIBUTIONS AND RELATED LOGARITHMIC INTEGRALS
The paper tackles two(2) problems related to the exponential distribution. The first concerns a detailed derivation of the minimum variance unbiased estimator of the scale parameter. The second focuses on the relationship of the expected value of the reciprocal of an exponential random variable which is shown to be equivalent to evaluating the logarithmic integral and the density of primes as found in the Prime Number Theorem. In the first problem, we showed that the minimum variance unbiased estimator of the scale parameter has a variance larger than the Cramer-Rao lower bound. In the second problem, we demonstrated that the expected value of the reciprocal of an exponential random variable also obtains the density of primes less or equal to a given large number x. The minimum variance unbiased estimator found in the first problem can then be utilized to find such an approximation to the density of primes for the second problem. The second problem provides a new way of viewing the problem of finding the density of primes less or equal to x.
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