Brownian motion, also known as Wiener process, is one of the most important and widely studied stochastic processes in both probability theory and mathematical physics. Originally observed in the erratic movement of pollen grains suspended in water by the botanist Robert Brown, it was later rigorously formalized by Norbert Wiener in the early 20th century. Brownian motion serves as a cornerstone in the modeling of various random phenomena, ranging from financial markets to the diffusion of particles in fluids. This manuscript provides a comprehensive overview of the key concepts, properties, and applications of Brownian motion. The exploration begins with a formal definition of Brownian motion and its fundamental properties, such as stationary independent increments and the Gaussian distribution of the process.

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Analytics is becoming critical in all part of our lives. Biostatistics has been a big driver of this analytics demand in the field of pharmaceuticals, biotech, health & medicine. Biostatistics (or biometry) is the application of statistics to a wide range of topics in biology. The science of biostatistics encompasses the design of biological experiments, especially in medicine, pharmacy, agriculture and fishery; the collection, summarization, and analysis of data from those experiments; and the interpretation of, and inference from, the results. A major branch of this is medical biostatistics,[1] which is exclusively concerned with medicine and health.