T-cell receptors (TCRs) play a crucial role in the immune system by recognizing and binding to specific antigens presented by infected or cancerous cells. Understanding the sequence patterns of TCRs is essential for developing targeted immune therapies and designing effective vaccines. Language models, such as auto-regressive transformers, offer a powerful solution to this problem by learning the probability distributions of TCR repertoires, enabling the generation of new TCR sequences that inherit the underlying patterns of the repertoire. We introduce TCR-GPT, a probabilistic model built on a decoder-only transformer architecture, designed to uncover and replicate sequence patterns in TCR repertoires. TCR-GPT demonstrates an accuracy of 0.953 in inferring sequence probability distributions measured by Pearson correlation coefficient. Furthermore, by leveraging Reinforcement Learning(RL), we adapted the distribution of TCR sequences to generate TCRs capable of recognizing specific peptides, offering significant potential for advancing targeted immune therapies and vaccine development. With the efficacy of RL, fine-tuned pretrained TCR-GPT models demonstrated the ability to produce TCR repertoires likely to bind specific peptides, illustrating RL's efficiency in enhancing the model's adaptability to the probability distributions of biologically relevant TCR sequences.