In
statistics and
physics,
Hodgson's paradox is the observation that the ratio of two
Normally distributed random variables has neither
mean nor
variance, and thus no well-defined
expectation. This appears to be inconsistent with conventional views of error estimation. The paradox is named for physicist R. T. Hodgson.
If
and
are normal variables with arbitrary mean and variance, then
has a
Cauchy distribution, which has no first moment.
The resolution of the paradox is to observe that random variables are never exactly Gaussian. The example Hodgson uses is that of the height of men: this cannot be Gaussian because heights cannot be negative (and the PDF for the normal distribution is positive for the whole real line).
Reference
* R. T. Hodgson. "The problem of being a normal deviate",
American Journal of Physics 47(12), December 1979.