  Considerable recent attention has focussed on the prospects to use the cosmic
microwave background (CMB) trispectrum to probe the physics of the early
universe. Here we evaluate the probability distribution function (PDF) for the
standard estimator tau_nle for the amplitude tau_nl of the CMB trispectrum both
for the null-hypothesis (i.e., for Gaussian maps with tau_nl = 0) and for maps
with a non-vanishing trispectrum (|tau_nl|>0). We find these PDFs to be highly
non-Gaussian in both cases. We also evaluate the variance with which the
trispectrum amplitude can be measured, <tau_nle^2>, as a function of its
underlying value, tau_nl. We find a strong dependence of this variance on
tau_nl. We also find that the variance does not, given the highly non-Gaussian
nature of the PDF, effectively characterize the distribution. Detailed
knowledge of these PDFs will therefore be imperative in order to properly
interpret the implications of any given trispectrum measurement. For example,
if a CMB experiment with a maximum multipole of lmax = 1500 (such as the Planck
satellite) measures tau_nle = 0 then at the 95% confidence our calculations
show that we can conclude tau_nl < 1005; assuming a Gaussian PDF but with the
correct tau_nl-dependent variance we would incorrectly conclude tau_nl < 4225;
further neglecting the tau_nl-dependence in the variance we would incorrectly
conclude tau_nl < 361.
