Maximum Likelihood and Unbiased Estimators

Hopefully, you can see that the Law of Large Numbers simplifies matters greatly when it comes to estimating an expected value or the mean of a distribution. Without knowledge of the distribution, or knowledge of the probabilities of the individual observations, we can nevertheless approximate the expected value and estimate the mean by calculating the average of a "large" sample from the population. In fact, we call the sample average the "maximum likelihood" estimator of the mean of the population. If it turns out that the expected value of the estimator is in fact equal to the parameter being estimated, then the estimator is said to be "unbiased." The sample average is an unbiased estimator of μ since the expected value of the sample average is also μ:

The practical effect of being unbiased is that as more and more observations are added to the sample, the expected value of the estimator becomes ever increasingly identical with the parameter being estimated. If there were a bias, the expected value might "wander off" with additional observations. ^[22]

Working the problem the other way, if the project manager knows expected value from a calculation using distributions and three-point estimates, then the project manager can deduce that a sample might contain the X_i. In fact, using Chebyshev's Inequality we find that the probability of an X_i straying very far from the mean, μ, goes down by the square of the distance from the mean. The probability that the absolute distance of sample value X_i from the population mean is greater than some distance, y, varies by 1/y²:

p(|X_i - μ | ≥ y) ≤ σ²/y²