It is known that at high doses ionizing radiation can cause cancer or leukemia. The functional relationship between cancer (leukemia) induction and received dose of ionizing radiation is still unknown, particularly in a low dose region. In this thesis atomic bomb survivors data are used to test two models, a linear threshold model for solid cancers and leukemia data and a liner-quadratic model for leukemia data only. Atomic bomb survivors data used in this thesis include data for stomach, lung, all solid cancers (all cancers excluding leukemia), and leukemia. Cancer and leukemia mortality rates and excess mortality rates are investigated as function of received dose using the standard Chi-square and a non-standard Monte Carlo simulation method. Using empirical data points one thousand simulated data sets were generated. Each simulated data set was fitted with a straight line, and intercept to dose axis, threshold, was calculated. This procedure gives one thousand threshold values. Statistical analysis of threshold values is used as a test of linear no-threshold and threshold models. In addition to a linear fit, a linear-quadratic fit was performed for leukemia data. In order to test a hormesis hypothesis Zero equivalent points ('ZEP') have been calculated. Upper threshold limits obtained by Monte Carlo simulation are 0.037 Sv and 0.061 Sv for all solid cancers, and 0.154 and 0.193 Sv for leukemia data sets. Investigation of mortality rates shows that the threshold and quadratic models do not fit data significantly better than the linear model.