  We present a determination of the distributions of photon spectral index and
gamma-ray flux - the so called LogN-LogS relation - for the 352 blazars
detected with a greater than approximately seven sigma detection threshold and
located above +/- 20 degrees Galactic latitude by the Large Area Telescope of
the Fermi Gamma-ray Space Telescope in its first year catalog. Because the flux
detection threshold depends on the photon index, the observed raw distributions
do not provide the true LogN-LogS counts or the true distribution of the photon
index. We use the non-parametric methods developed by Efron and Petrosian to
reconstruct the intrinsic distributions from the observed ones which account
for the data truncations introduced by observational bias and includes the
effects of the possible correlation between the two variables. We demonstrate
the robustness of our procedures using a simulated data set of blazars and then
apply these to the real data and find that for the population as a whole the
intrinsic flux distribution can be represented by a broken power law with high
and low indexes of -2.37 +/- 0.13 and -1.70 +/- 0.26, respectively, and the
intrinsic photon index distribution can be represented by a Gaussian with mean
of 2.41 +/- 0.13 and width of 0.25 +/- 0.03. We also find the intrinsic
distributions for the sub-populations of BL Lac and FSRQs type blazars
separately. We then calculate the contribution of Fermi blazars to the diffuse
extragalactic gamma-ray background radiation. Under the assumption that the
flux distribution of blazars continues to arbitrarily low fluxes, we calculate
the best fit contribution of all blazars to the total extragalactic gamma-ray
output to be 60%, with a large uncertainty.
