  The Fisher metric on a manifold of probability distributions is usually
treated as a metric on the tangent bundle. In this paper, we focus on the
metric on the cotangent bundle induced from the Fisher metric with calling it
the Fisher co-metric. We show that the Fisher co-metric can be defined directly
without going through the Fisher metric by establishing a natural
correspondence between cotangent vectors and random variables. This definition
clarifies a close relation between the Fisher co-metric and the
variance/covariance of random variables, whereby the Cram\'{e}r-Rao inequality
is trivialized. We also discuss the monotonicity and the invariance of the
Fisher co-metric with respect to Markov maps, and present a theorem
characterizing the co-metric by the invariance, which can be regarded as a
cotangent version of \v{C}encov's characterization theorem for the Fisher
metric. The obtained theorem can also viewed as giving a characterization of
the variance/covariance.
