The Teacher's Corner: A Note on the Expectation of the Reciprocal and Square Root of a Random Variable

In The American Statistician, February, 1966 (vol. 20, Number 1), [1] a well-known result, viz., the expectation of a positive random variable with a finite mean such that it is not identically equal to zero, is greater than the reciprocal of its expectation has been proved by a new simple method different from the conventional method depending on Schwarz's inequality [2] (which is also quite simple). There is however a still shorter method which the authors have been employing in statistics classes which might bear repetition although it should be well-known. It is as follows: