We present a determination of the distributions of gamma-ray flux - the so called LogN-LogS relation - and photon spectral index 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 among 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 of slopes -2.37 +/- 0.13 and -1.70 +/- 0.26, and the intrinsic photon index distribution can be represented by a Gaussian with mean 2.41 +/- 0.13 and 1 sigma 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 blazars to the diffuse cosmic gamma-ray background radiation to be 28% +/- 19%. Show Partial Abstract