Parameter distribution estimation in first order ODE

With development of new technologies applied to biological experiments, more and more data are generated every day. To make predictions in biological systems, mathematical modeling plays a critical role. Ordinary differential equations (ODEs) contribute to a large portion in mathematical modeling. In which parameters are inevitable. Noise is intrinsic in all experiments. Therefore, to think of parameters as statistical distributions is a realistic treatment. In this paper, we discuss in a 1st order ODE common in biological systems, how to calculate parameter distribution analytically according to the experimentally observed output assumed to be normal distribution. Conditions on when parameter can be correctly estimated are elucidated.