Most approaches to sentiment analysis requires a sentiment lexicon in order to automatically predict sentiment or opinion in a text. The lexicon is generated by selecting words and assigning scores to the words, and the performance the sentiment analysis depends on the quality of the assigned scores. This paper addresses an aspect of sentiment lexicon generation that has been overlooked so far; namely that the most appropriate score assigned to a word in the lexicon is dependent on the domain. The common practice, on the contrary, is that the same lexicon is used without adjustments across different domains ignoring the fact that the scores are normally highly sensitive to the domain. Consequently, the same lexicon might perform well on a single domain while performing poorly on another domain, unless some score adjustment is performed. In this paper, we advocate that a sentiment lexicon needs some further adjustments in order to perform well in a specific domain. In order to cope with these domain specific adjustments, we adopt a stochastic formulation of the sentiment score assignment problem instead of the classical deterministic formulation. Thus, viewing a sentiment score as a stochastic variable permits us to accommodate to the domain specific adjustments. Experimental results demonstrate the feasibility of our approach and its superiority to generic lexicons without domain adjustments.