Detecting anomalies in streaming data presents unique challenges due to the continuous and dynamic nature of the data. Traditional anomaly detection techniques are often unsuitable for real-time environments where adaptability and scalability are paramount. In this paper, we propose a novel window-based anomaly detection algorithm that leverages the Bayesian ARIMA (AutoRegressive Integrated Moving Average) model for real-time streaming data. Our approach dynamically computes threshold values within sliding windows, allowing for the identification of local and global anomalies in an adaptive manner. The main contribution of our work is the development of a streaming Bayesian ARIMA model for real-time anomaly detection in continuous time-series data, effectively bridging the gap in leveraging statistical models for dynamic data streams. Extensive experiments on benchmark streaming datasets demonstrate that our algorithm outperforms the state-of-the-art algorithms in terms of AUC-ROC values.