This document discusses statistical concepts including the expectation value of the average, variation, and covariance for a normal distribution. It defines an unbiased estimator as a function whose expectation value is equal to the true average, variation, or covariance. It also cautions about making inferences based on limited information through an anecdote about statisticians on a plane with failing engines.

1. APPLIED STATISTICS Torture the data long enough and they will confess to anything.

2. Expectation Values of Average, Variation and Covariance Estimators of Average, Variation and Covariance for Normal Distribution CONTENTS

3. EXPECTATION VALUE OF AVERAGE – GENERAL DEFINITION FOR CONTINUOUS DISTRIBUTION, F

4. Normal Distribution True variation variable True average Then there is the man who drowned crossing a stream with an average depth of six inches.

5. EXPECTATION VALUE OF AVERAGE FOR A NORMAL DISTRIBUTION Statistics say that one out of every four Americans is suffering from some form of mental illness. Think of your three best friends. If they're okay, then it's you.

6. DEFINITION OF UNBIASED ESTIMATOR An unbiased estimator of the average is a function whose expectation value is the true average. An unbiased estimator of the variation is a function whose expectation value is the true variation. An unbiased estimator of the covariance is a function whose expectation value is the true covariance.

7. ESTIMATOR OF AVERAGE

8. EXPECTATION VALUE OF VARIATION General definition is

9. ESTIMATOR OF VARIATION

10. THE FALLACY OF INFERENCE Two statisticians were flying from L.A. to New York. About an hour into the flight, the pilot announced, "Unfortunately, we have lost an engine, but don't worry: There are three engines left. However, instead of five hours, it will take seven hours to get to New York." A little later, he told the passengers that a second engine had failed. "But we still have two engines left. We're still fine, except now it will take ten hours to get to New York." Somewhat later, the pilot again came on the intercom and announced that a third engine had died. "But never fear, because this plane can fly on a single engine. Of course, it will now take 18 hours to get to New York." At this point, one statistician turned to another and said, "Gee, I hope we don't lose that last engine, or we'll be up here forever!"

11. EXPECTATION VALUE and ESTIMATOR OF COVARIANCE