Suppose X takes the value 1 with probability 0.1 and the value 10 with probability 0.9.
The harmonic mean is another kind of average (like the geometric mean), defined by H(X) = 1/E(1/X) (the reciprocal of the expectation of the reciprocal). Like the geometric mean, the harmonic mean is only defined for random variables that take on positive values, and it is always less than the expectation. Compute the harmonic mean of the following random variables and check that it is indeed less than the expectation.