Lymphocytes, comprising of B and T cells, are important members of the adaptive immune system of vertebrates that play a crucial role in defending against harmful pathogens. They are equipped with receptors capable of recognising specific antigens. After activation, they proliferate to form an exponentially growing clone army. Eventually, those cells cease to divide and then largely die over a period of weeks, but leave a small number of cells, called memory cells, that can rapidly respond to any repeated infection. To study such non-linear population dynamics, experimental systems have been designed that generate data at the level of populations, families and single cells to elucidate underlying mechanisms that regulate expansion, cessation, and contraction of cell numbers. In this thesis, we report on the development of a novel stochastic model of cellular population dynamics, based on Hawkins et al. (2007a), that accounts for experimentally observed correlation structure within family members. In particular, the inheritance of cell division, cessation, and death times within a stochastic model framework considered, and their impact on cell population dynamics are investigated. Model assumptions are informed by datasets from time-lapse microscopy experiments and statistically tested within the Bayesian framework. Consequences of the dependencies are demonstrated with family trees generated by a Monte-Carlo simulation. To assess the model's ability to extract meaningful inferences from population-level data, we design an optimisation strategy to estimate model parameters and investigate its accuracy and precision for a given dataset from in vitro murine system. With the analysis pipeline, the model is applied to both in vitro murine and human lymphocyte populations to test hypotheses and draw meaningful biological conclusions. For instance, we demonstrate signal integration for T cells from transgenic mice as a linear sum in a time domain, and as a result, the model successfully recapitulates the data. Lastly, we extend the remit of the stochastic modelling framework by exploring mechanisms of B cell differentiation to antibody-secreting cells and their class switching to different isotypes. A simple probabilistic model that captures molecular changes within these cells sheds light on the process of determining the types of antibodies to produce and predicting the magnitude associated with them.