Positron emission tomography (PET) can provide in vivo, quantitative and functional information for diagnosis; however, PET image quality depends highly on a reconstruction algorithm. Iterative algorithms, such as the maximum likelihood expectation maximization (MLEM) algorithm, are rapidly becoming the standards for image reconstruction in emission-computed tomography. The conventional MLEM algorithm utilized the Poisson model in its system matrix, which is no longer valid for delay-subtraction of randomly corrected data. The aim of this study is to overcome this problem. The maximum likelihood estimation using the expectation maximum algorithm (MLE-EM) is adopted and modified to reconstruct microPET images using random correction from joint prompt and delay sinograms; this reconstruction method is called PDEM. The proposed joint Poisson model preserves Poisson properties without increasing the variance (noise) associated with random correction. The work here is an initial application/demonstration without applied normalization, scattering, attenuation, and arc correction. The coefficients of variation (CV) and full width at half-maximum (FWHM) values were utilized to compare the quality of reconstructed microPET images of physical phantoms acquired by filtered backprojection (FBP), ordered subsets-expected maximum (OSEM) and PDEM approaches. Experimental and simulated results demonstrate that the proposed PDEM produces better image quality than the FBP and OSEM approaches.