  We present results of correlation studies, examining the association between
the peak of the nu F_nu spectrum of gamma ray bursts, E_p, with the burst's
energy fluence and photon peak flux. We discuss methods to account for data
truncation in E_p and fluence or flux when performing the correlation analyses.
  However, because bursts near the detector threshold are not usually able to
provide reliable spectral parameters, we focus on results for the brightest
bursts in which we can better understand the selection effects relevant to E_p
and burst strength.
  We find that there is a strong correlation between total fluence and E_p. We
discuss these results in terms of both cosmological and intrinsic effects.
  In particular, we show that for realistic distributions of the burst
parameters, cosmological expansion alone cannot account for the correlation
between E_p and total fluence; the observed correlation is likely a result of
an intrinsic relation between the burst rest-frame peak energy and the total
radiated energy. We investigate this latter scenario in the context of
synchrotron radiation from external and internal shock models of GRBs. We find
that the internal shock model is consistent with our interpretation of the
correlation, while the external shock model cannot easily explain this
intrinsic relation between peak energy and burst radiated energy.
